You are viewing a plain text version of this content. The canonical link for it is here.
Posted to commits@commons.apache.org by ah...@apache.org on 2021/11/25 16:17:43 UTC
[commons-statistics] branch master updated: Update normal distribution pdf to compute directly.
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a commit to branch master
in repository https://gitbox.apache.org/repos/asf/commons-statistics.git
The following commit(s) were added to refs/heads/master by this push:
new d731ea4 Update normal distribution pdf to compute directly.
d731ea4 is described below
commit d731ea4530feb70cf66801aa84e4b813493cecc7
Author: aherbert <ah...@apache.org>
AuthorDate: Thu Nov 25 16:16:28 2021 +0000
Update normal distribution pdf to compute directly.
Use of exp(logDensity) can suffer errors as x increases in magnitude for
the standard normal distribution (mu=0, sd=1).
Approximate errors:
x ulp
3 3
6 10
12 >40
Added extended precision test data for the standard normal distribution
computed using maxima. ULP errors are below 2 for x <= 12 when computing
the PDF directly.
Added extended precision computation of x * sqrt(2 * pi) for use in the
PDF computation for normal distributions with a mean other than 1.
This computes more accurately than x * Math.sqrt(2 * Math.PI), or x *
SQRT2PI if using a pre-computed constant.
---
.../statistics/distribution/ExtendedPrecision.java | 68 +
.../distribution/NormalDistribution.java | 10 +-
.../distribution/ExtendedPrecisionTest.java | 57 +-
.../distribution/NormalDistributionTest.java | 19 +-
.../commons/statistics/distribution/normpdf.csv | 1564 ++++++++++++++++++++
.../commons/statistics/distribution/xsqrt2pi.csv | 527 +++++++
6 files changed, 2233 insertions(+), 12 deletions(-)
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
index 7fa3c93..c12e7d0 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
@@ -16,6 +16,8 @@
*/
package org.apache.commons.statistics.distribution;
+import java.math.BigDecimal;
+
/**
* Computes extended precision floating-point operations.
*
@@ -27,6 +29,8 @@ package org.apache.commons.statistics.distribution;
* <p>Adapted from {@code org.apache.commons.numbers.core.ExtendedPrecion}.
*/
final class ExtendedPrecision {
+ /** sqrt(2 pi). Computed to 64-digits. */
+ private static final String SQRT_TWO_PI = "2.506628274631000502415765284811045253006986740609938316629923576";
/**
* The multiplier used to split the double value into high and low parts. From
* Dekker (1971): "The constant should be chosen equal to 2^(p - p/2) + 1,
@@ -42,11 +46,75 @@ final class ExtendedPrecision {
private static final double SCALE_UP = 0x1.0p600;
/** Scale down by 2^600. */
private static final double SCALE_DOWN = 0x1.0p-600;
+ /** sqrt(2 pi) as a double. */
+ private static final double SQRT2PI;
+ /** Upper bits of sqrt(2 pi). */
+ private static final double SQRT2PI_H;
+ /** Lower bits of sqrt(2 pi). */
+ private static final double SQRT2PI_L;
+ /** Round-off from sqrt(2 pi) as a double. */
+ private static final double SQRT2PI_R;
+
+ static {
+ // Initialise constants
+ final BigDecimal sqrt2pi = new BigDecimal(SQRT_TWO_PI);
+
+ // Use a 106-bit number as:
+ // (SQRT2PI, SQRT2PI_R)
+ SQRT2PI = sqrt2pi.doubleValue();
+ SQRT2PI_R = sqrt2pi.subtract(new BigDecimal(SQRT2PI)).doubleValue();
+
+ // Split the upper 53-bits for extended precision multiplication
+ SQRT2PI_H = highPartUnscaled(SQRT2PI);
+ SQRT2PI_L = SQRT2PI - SQRT2PI_H;
+ }
/** No instances. */
private ExtendedPrecision() {}
/**
+ * Multiply the term by sqrt(2 pi).
+ *
+ * @param x Value (assumed to be positive)
+ * @return x * sqrt(2 pi)
+ */
+ static double xsqrt2pi(double x) {
+ // Note: Do not convert x to absolute for this use case
+ if (x > BIG) {
+ if (x == Double.POSITIVE_INFINITY) {
+ return Double.POSITIVE_INFINITY;
+ }
+ return computeXsqrt2pi(x * SCALE_DOWN) * SCALE_UP;
+ } else if (x < SMALL) {
+ // Note: Ignore possible zero for this use case
+ return computeXsqrt2pi(x * SCALE_UP) * SCALE_DOWN;
+ } else {
+ return computeXsqrt2pi(x);
+ }
+ }
+
+ /**
+ * Compute {@code a * sqrt(2 * pi)}.
+ *
+ * @param a Value
+ * @return the result
+ */
+ private static double computeXsqrt2pi(double a) {
+ // Split the number
+ final double ha = highPartUnscaled(a);
+ final double la = a - ha;
+
+ // Extended precision product with sqrt(2 * pi)
+ final double x = a * SQRT2PI;
+ final double xx = productLow(ha, la, SQRT2PI_H, SQRT2PI_L, x);
+
+ // Add the term a multiplied by the round-off from sqrt(2 * pi)
+ // result = a * (SQRT2PI + SQRT2PI_R)
+ // Sum from small to high
+ return a * SQRT2PI_R + xx + x;
+ }
+
+ /**
* Compute {@code sqrt(2 * x * x)}.
*
* <p>The result is computed using a high precision computation of
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
index fd0be84..2848dd4 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
@@ -49,6 +49,10 @@ public final class NormalDistribution extends AbstractContinuousDistribution {
* differences as the error function computes close to 0 in the extreme tail.
*/
private final double sdSqrt2;
+ /**
+ * Standard deviation multiplied by sqrt(2 pi). Computed to high precision.
+ */
+ private final double sdSqrt2pi;
/**
* @param mean Mean for this distribution.
@@ -62,6 +66,8 @@ public final class NormalDistribution extends AbstractContinuousDistribution {
// Minimise rounding error by computing sqrt(2 * sd * sd) exactly.
// Compute using extended precision with care to avoid over/underflow.
sdSqrt2 = ExtendedPrecision.sqrt2xx(sd);
+ // Compute sd * sqrt(2 * pi)
+ sdSqrt2pi = ExtendedPrecision.xsqrt2pi(sd);
}
/**
@@ -93,7 +99,9 @@ public final class NormalDistribution extends AbstractContinuousDistribution {
/** {@inheritDoc} */
@Override
public double density(double x) {
- return Math.exp(logDensity(x));
+ final double x0 = x - mean;
+ final double x1 = x0 / standardDeviation;
+ return Math.exp(-0.5 * x1 * x1) / sdSqrt2pi;
}
/** {@inheritDoc} */
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
index 3c6398a..9378cdf 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
@@ -25,6 +25,7 @@ import org.junit.jupiter.api.Test;
import org.junit.jupiter.api.TestMethodOrder;
import org.junit.jupiter.params.ParameterizedTest;
import org.junit.jupiter.params.provider.CsvFileSource;
+import org.junit.jupiter.params.provider.ValueSource;
/**
* Test for {@link ExtendedPrecision}.
@@ -33,10 +34,14 @@ import org.junit.jupiter.params.provider.CsvFileSource;
class ExtendedPrecisionTest {
/** sqrt(2). */
private static final double ROOT2 = Math.sqrt(2.0);
- /** The sum of the squared ULP error for the first standard computation. */
- private static final RMS RMS1 = new RMS();
- /** The sum of the squared ULP error for the second standard computation. */
- private static final RMS RMS2 = new RMS();
+ /** sqrt(2 * pi) to 64 digits. This is 1 ULP different from Math.sqrt(2 * Math.PI). */
+ private static final double ROOT2PI = 2.506628274631000502415765284811045253006986740609938316629923576;
+ /** The sum of the squared ULP error for the first standard computation for sqrt(2 * x * x). */
+ private static final RMS SQRT2XX_RMS1 = new RMS();
+ /** The sum of the squared ULP error for the second standard computation for sqrt(2 * x * x). */
+ private static final RMS SQRT2XX_RMS2 = new RMS();
+ /** The sum of the squared ULP error for the first computation for x * sqrt(2 pi). */
+ private static final RMS XSQRT2PI_RMS = new RMS();
/**
* Class to compute the root mean squared error (RMS).
@@ -120,7 +125,7 @@ class ExtendedPrecisionTest {
/**
* Test the extended precision {@code sqrt(2 * x * x)}. The expected result
- * is an exact computation. For comparison ulp errors are collected for
+ * is an extended precision computation. For comparison ulp errors are collected for
* two standard precision computations.
*
* @param x Value x
@@ -133,23 +138,55 @@ class ExtendedPrecisionTest {
final double e = expected.doubleValue();
Assertions.assertEquals(e, ExtendedPrecision.sqrt2xx(x));
// Compute error for the standard computations
- addError(Math.sqrt(2 * x * x), expected, e, RMS1);
- addError(x * ROOT2, expected, e, RMS2);
+ addError(Math.sqrt(2 * x * x), expected, e, SQRT2XX_RMS1);
+ addError(x * ROOT2, expected, e, SQRT2XX_RMS2);
}
@Test
void testSqrt2xxStandardPrecision1() {
// Typical result: max 0.7780 rms 0.2144
- assertSqrt2xxStandardPrecision(RMS1, 0.9, 0.3);
+ assertPrecision(SQRT2XX_RMS1, 0.9, 0.3);
}
@Test
void testSqrt2xxStandardPrecision2() {
// Typical result: max 1.0598 rms 0.4781
- assertSqrt2xxStandardPrecision(RMS2, 1.3, 0.6);
+ assertPrecision(SQRT2XX_RMS2, 1.3, 0.6);
}
- private static void assertSqrt2xxStandardPrecision(RMS rms, double maxError, double rmsError) {
+ @ParameterizedTest
+ @ValueSource(doubles = {0, 1, Double.MAX_VALUE, Double.POSITIVE_INFINITY, Double.NaN})
+ void testXsqrt2piEdgeCases(double x) {
+ final double expected = x * ROOT2PI;
+ final double actual = ExtendedPrecision.xsqrt2pi(x);
+ Assertions.assertEquals(expected, actual, 1e-15);
+ }
+
+ /**
+ * Test the extended precision {@code x * sqrt(2 * pi)}. The expected result
+ * is an extended precision computation. For comparison ulp errors are collected for
+ * a standard computation.
+ *
+ * @param x Value x
+ * @param expected Expected result of {@code x * sqrt(2 * pi)}.
+ */
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "xsqrt2pi.csv")
+ void testXsqrt2pi(double x, BigDecimal expected) {
+ final double e = expected.doubleValue();
+ Assertions.assertEquals(e, ExtendedPrecision.xsqrt2pi(x));
+ // Compute error for the standard computation
+ addError(x * ROOT2PI, expected, e, XSQRT2PI_RMS);
+ }
+
+ @Test
+ void testXsqrt2piPrecision() {
+ // Typical result: max 1.1397 rms 0.5368
+ assertPrecision(XSQRT2PI_RMS, 1.2, 0.6);
+ }
+
+ private static void assertPrecision(RMS rms, double maxError, double rmsError) {
Assertions.assertTrue(rms.getMax() < maxError, () -> "max error: " + rms.getMax());
Assertions.assertTrue(rms.getRMS() < rmsError, () -> "rms error: " + rms.getRMS());
}
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
index fe16072..4f8e42b 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
@@ -17,14 +17,21 @@
package org.apache.commons.statistics.distribution;
+import java.math.BigDecimal;
import org.junit.jupiter.api.Assertions;
import org.junit.jupiter.api.Test;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.CsvFileSource;
/**
* Test cases for {@link NormalDistribution}.
* Extends {@link BaseContinuousDistributionTest}. See javadoc of that class for details.
*/
class NormalDistributionTest extends BaseContinuousDistributionTest {
+ /** A standard normal distribution used for calculations.
+ * This is immutable and thread-safe and can be used across instances. */
+ private static final NormalDistribution STANDARD_NORMAL = NormalDistribution.of(0, 1);
+
@Override
ContinuousDistribution makeDistribution(Object... parameters) {
final double mean = (Double) parameters[0];
@@ -56,7 +63,7 @@ class NormalDistributionTest extends BaseContinuousDistributionTest {
protected double getHighPrecisionRelativeTolerance() {
// Tests are limited by the survival probability.
// Tolerance is 1.6653345369377348E-14.
- // This is lowest achieved with various implementations of the
+ // This is the lowest achieved with various implementations of the
// survival function against high precision reference data.
// It requires computing the factor sqrt(2 * sd * sd) exactly.
return 75 * RELATIVE_EPS;
@@ -169,4 +176,14 @@ class NormalDistributionTest extends BaseContinuousDistributionTest {
Assertions.assertEquals(x, x0, Math.abs(x) * 1e-11, () -> "CDF = " + cdf);
}
}
+
+ @ParameterizedTest
+ @CsvFileSource(resources = "normpdf.csv")
+ void testPDF(double x, BigDecimal expected) {
+ final double e = expected.doubleValue();
+ final double a = STANDARD_NORMAL.density(x);
+ // Require high precision. Currently this does not work at 1 ULP.
+ Assertions.assertEquals(e, a, Math.ulp(e) * 2,
+ () -> "ULP error: " + expected.subtract(new BigDecimal(a)).doubleValue() / Math.ulp(e));
+ }
}
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf.csv
new file mode 100644
index 0000000..d012c32
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf.csv
@@ -0,0 +1,1564 @@
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements. See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+# High-precision test data for the normal distribution PDF
+# Computed using the following maxima script:
+#
+# fpprec : 128;
+# sqrt2pi : bfloat(2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266116009972665203879644866323618);
+# f(x) := exp(-0.5 * bfloat(x) * bfloat(x)) / sqrt2pi;
+# str(x) := ssubst("e","b",string(x));
+#
+# out : openw("normpdf.csv");
+# x : float(makelist(i / 128, i, 1, 128 * 12));
+# for i in x do printf(out, "~f, ~a~%", i, str(f(i))), fpprintprec : 30;
+# close(out);
+
+0.0078125, 3.98930105834993218489256026977e-1
+0.015625, 3.98893584364825255305769353747e-1
+0.0234375, 3.98832722677699497553511544723e-1
+0.03125, 3.98747531915966650955902402398e-1
+0.0390625, 3.98638027674157618347172884114e-1
+0.046875, 3.98504229994225854657585918961e-1
+0.0546875, 3.98346163359433879195504302147e-1
+0.0625, 3.98163856686886555927772779149e-1
+0.0703125, 3.97957343318714329694085544133e-1
+0.078125, 3.97726661011910181775267782946e-1
+0.0859375, 3.97471851926824641654316765421e-1
+0.09375, 3.97192962614323762848615803325e-1
+0.1015625, 3.96890044001615539039540061316e-1
+0.109375, 3.96563151376750802071108009751e-1
+0.1171875, 3.96212344371805205423799326526e-1
+0.125, 3.95837686944749455186780064425e-1
+0.1328125, 3.95439247360015505047591480732e-1
+0.140625, 3.95017098167766982091536663736e-1
+0.1484375, 3.94571316181882655955029706266e-1
+0.15625, 3.9410198245666230481256841677e-1
+0.1640625, 3.93609182262264867503483705127e-1
+0.171875, 3.93093005058889301533347615858e-1
+0.1796875, 3.92553544469709091430629065661e-1
+0.1875, 3.9199089825257187072040145004e-1
+0.1953125, 3.91405168270476133316131858756e-1
+0.203125, 3.90796460460837516154474120183e-1
+0.2109375, 3.90164884803557634137535773546e-1
+0.21875, 3.89510555287908940637787548673e-1
+0.2265625, 3.88833589878249571702806046894e-1
+0.234375, 3.88134110478582609415404373208e-1
+0.2421875, 3.87412242895974669369438709035e-1
+0.25, 3.8666811680284917866787714197e-1
+0.2578125, 3.85901865698170163998100747182e-1
+0.265625, 3.85113626867532813955872102701e-1
+0.2734375, 3.8430354134217751564557243698e-1
+0.28125, 3.83471753856944492457560819503e-1
+0.2890625, 3.82618412807186587597137032697e-1
+0.296875, 3.81743670204658146202922093077e-1
+0.3046875, 3.80847681632398347541002311281e-1
+0.3125, 3.79930606198627727596701545322e-1
+0.3203125, 3.78992606489677011216550415729e-1
+0.328125, 3.7803384852196774159363502283e-1
+0.3359375, 3.77054501693064553161390761063e-1
+0.34375, 3.76054738731819281692158519804e-1
+0.3515625, 3.75034735647627442422379220553e-1
+0.359375, 3.73994671678817933188039558209e-1
+0.3671875, 3.72934729240197134700789940605e-1
+0.375, 3.71855093869768884083032899794e-1
+0.3828125, 3.70755954174652090472407539617e-1
+0.390625, 3.69637501776218042772909121598e-1
+0.3984375, 3.68499931254469729349141245025e-1
+0.40625, 3.67343440091685747517043569978e-1
+0.4140625, 3.66168228615351626971452584641e-1
+0.421875, 3.64974499940401625708108291155e-1
+0.4296875, 3.63762459910794279452822095646e-1
+0.4375, 3.62532317040445196018650286353e-1
+0.4453125, 3.61284282453540784295859663501e-1
+0.453125, 3.60018569824256793669647935261e-1
+0.4609375, 3.58735395315905713495068579133e-1
+0.46875, 3.57434977519537243783157833581e-1
+0.4765625, 3.56117537392016197420303505413e-1
+0.484375, 3.54783298193602331015552849498e-1
+0.4921875, 3.53432485425056725816639069182e-1
+0.5, 3.52065326764299452031503580257e-1
+0.5078125, 3.50682052002643349322164215564e-1
+0.515625, 3.49282892980628843193740294011e-1
+0.5234375, 3.47868083523484791482735395792e-1
+0.53125, 3.46437859376240417162339980131e-1
+0.5390625, 3.4499245813851343324315771766e-1
+0.546875, 3.43532119198999502677518223561e-1
+0.5546875, 3.42057083669688200904034345134e-1
+0.5625, 3.40567594319830661033346866856e-1
+0.5703125, 3.39063895509684081720503892987e-1
+0.578125, 3.37546233124058265545894235456e-1
+0.5859375, 3.36014854505689331294093976308e-1
+0.59375, 3.34470008388465706944574774228e-1
+0.6015625, 3.32911944830531461543249524082e-1
+0.609375, 3.31340915147291973489609784645e-1
+0.6171875, 3.29757171844346860237995107405e-1
+0.625, 3.28160968550375010067433470377e-1
+0.6328125, 3.26552559949996460523367864805e-1
+0.640625, 3.24932201716635760483958273674e-1
+0.6484375, 3.23300150445411333667597321592e-1
+0.65625, 3.21656663586075230897325666915e-1
+0.6640625, 3.20001999376027516698839324184e-1
+0.671875, 3.18336416773429382964824983401e-1
+0.6796875, 3.16660175390438918608620539468e-1
+0.6875, 3.14973535426593289499832387258e-1
+0.6953125, 3.13276757602360897674553526805e-1
+0.703125, 3.11570103092886892999940114284e-1
+0.7109375, 3.0985383346195520430942522323e-1
+0.71875, 3.08128210596190040678529261002e-1
+0.7265625, 3.06393496639519587155123157647e-1
+0.734375, 3.04649953927924383070329698355e-1
+0.7421875, 3.02897844924492625220239680013e-1
+0.75, 3.0113743215480438291236516817e-1
+0.7578125, 2.99368978142666447307054151386e-1
+0.765625, 2.97592745346219263850305942587e-1
+0.7734375, 2.95808996094437114092307948025e-1
+0.78125, 2.94017992524042422021551477544e-1
+0.7890625, 2.92219996516854760427640672474e-1
+0.796875, 2.90415269637594824950857545578e-1
+0.8046875, 2.88604073072163327600904031134e-1
+0.8125, 2.86786667566414437852300565901e-1
+0.8203125, 2.84963313365443068174377415763e-1
+0.828125, 2.83134270153404962257581596265e-1
+0.8359375, 2.81299796993888198485932772706e-1
+0.84375, 2.79460152270854368611778438644e-1
+0.8515625, 2.77615593630167332350118323977e-1
+0.859375, 2.75766377921727082964825132274e-1
+0.8671875, 2.73912761142225887109580268395e-1
+0.875, 2.72054998378543484455951439596e-1
+0.8828125, 2.70193343751797749235454198052e-1
+0.890625, 2.68328050362066826989182411892e-1
+0.8984375, 2.66459370233798365806836799173e-1
+0.90625, 2.64587554261921062397374480121e-1
+0.9140625, 2.6271285215867333971799174259e-1
+0.921875, 2.60835512401163564849802524549e-1
+0.9296875, 2.58955782179675803601395054407e-1
+0.9375, 2.57073907346734692200213657996e-1
+0.9453125, 2.55190132366942586651787593169e-1
+0.953125, 2.53304700267601727163995221473e-1
+0.9609375, 2.51417852590133728703832789261e-1
+0.96875, 2.49529829342308279533644104441e-1
+0.9765625, 2.47640868951292497718448825358e-1
+0.984375, 2.4575120821753196136159765251e-1
+0.9921875, 2.4386108226947399196775450421e-1
+1.0, 2.41970724519143332104820734117e-1
+1.0078125, 2.40080366618579918693760726969e-1
+1.015625, 2.38190238417148012050308105313e-1
+1.0234375, 2.36300567919725498487071796154e-1
+1.03125, 2.34411581245781741109204052558e-1
+1.0390625, 2.32523502589351909650701294594e-1
+1.046875, 2.30636554179915276049171236465e-1
+1.0546875, 2.28750956244184518190375725897e-1
+1.0625, 2.26866926968812630114028440387e-1
+1.0703125, 2.24984682464023593201143174507e-1
+1.078125, 2.23104436728172519700472228632e-1
+1.0859375, 2.21226401613240537634710141785e-1
+1.09375, 2.19350786791269244891175021764e-1
+1.1015625, 2.17477799721739120379034920765e-1
+1.109375, 2.15607645619895841755510626871e-1
+1.1171875, 2.13740527426028022613692219958e-1
+1.125, 2.11876645775699447408503218293e-1
+1.1328125, 2.1001619897093844999567225658e-1
+1.140625, 2.08159382952386651688836719609e-1
+1.1484375, 2.06306391272408847416266226448e-1
+1.15625, 2.04457415069165404091854567135e-1
+1.1640625, 2.02612643041648113912111774932e-1
+1.171875, 2.00772261425680027155338328901e-1
+1.1796875, 1.98936453970879374390642051986e-1
+1.1875, 1.97105401918587276998742427457e-1
+1.1953125, 1.9527928398075853775539427786e-1
+1.203125, 1.9345827631981440011947827178e-1
+1.2109375, 1.91642552529455765984913954416e-1
+1.21875, 1.89832283616434967177870772879e-1
+1.2265625, 1.88027637983283796083276267404e-1
+1.234375, 1.86228781411995115637937172483e-1
+1.2421875, 1.84435877048654988697811072878e-1
+1.25, 1.82649085389021891635660339492e-1
+1.2578125, 1.80868564265049207109442423672e-1
+1.265625, 1.79094468832346826413625378685e-1
+1.2734375, 1.773269515585773328327197463e-1
+1.28125, 1.75566162212781884101460421486e-1
+1.2890625, 1.73812247855630564577196003056e-1
+1.296875, 1.72065352830591636180212953488e-1
+1.3046875, 1.70325618756013781685084209396e-1
+1.3125, 1.68593184518115104673878151911e-1
+1.3203125, 1.66868186264872327508397134933e-1
+1.328125, 1.6515075740080331215672050106e-1
+1.3359375, 1.63441028582635718727347637081e-1
+1.34375, 1.61739127715854313225250208769e-1
+1.3515625, 1.60045179952119139446145482095e-1
+1.359375, 1.58359307687546480161195326844e-1
+1.3671875, 1.5668163056184424990191362139e-1
+1.375, 1.55012265458293185817012678579e-1
+1.3828125, 1.5335132650456493431680801567e-1
+1.390625, 1.5169892507436786961908959319e-1
+1.3984375, 1.50055169789911225930409420208e-1
+1.40625, 1.48420166525177877900785579536e-1
+1.4140625, 1.46794018409995864235051802715e-1
+1.421875, 1.45176825834898516982591536251e-1
+1.4296875, 1.43568686456762834106038753284e-1
+1.4375, 1.41969695205215515490731433324e-1
+1.4453125, 1.40379944289795872637295148858e-1
+1.453125, 1.38799523207864619911771590882e-1
+1.4609375, 1.37228518753247360438311894297e-1
+1.46875, 1.35667015025601392530849197946e-1
+1.4765625, 1.34115093440494282989710638772e-1
+1.484375, 1.32572832740182481649370002971e-1
+1.4921875, 1.31040309005078087262250743555e-1
+1.5, 1.29517595665891718143714040228e-1
+1.5078125, 1.2800476351643929198348541669e-1
+1.515625, 1.2650188072710037784254628999e-1
+1.5234375, 1.25009012858915649591143727341e-1
+1.53125, 1.235262228783108438877523048e-1
+1.5390625, 1.22053571172434507230817999468e-1
+1.546875, 1.20591115565096705610814100605e-1
+1.5546875, 1.19138911333295766821125991274e-1
+1.5625, 1.17697011224320029519901970224e-1
+1.5703125, 1.1626546547341148463444344291e-1
+1.578125, 1.14844321821978113624052036073e-1
+1.5859375, 1.13433625536341654421581722943e-1
+1.59375, 1.12033419427007459509403331405e-1
+1.6015625, 1.10643743868443051499363608396e-1
+1.609375, 1.09264636819351929722124359751e-1
+1.6171875, 1.07896133843429136628822071663e-1
+1.625, 1.06538268130585055203513172216e-1
+1.6328125, 1.05191070518623878011065299421e-1
+1.640625, 1.03854569515363164891291886599e-1
+1.6484375, 1.02528791321180889582137851581e-1
+1.65625, 1.01213759851976365635291625934e-1
+1.6640625, 9.9909496762531438796250632494e-2
+1.671875, 9.861602147025833647406855844e-2
+1.6796875, 9.73333511793205749372603557493e-2
+1.6875, 9.6061500905113341352259340499e-2
+1.6953125, 9.48004834990897906372584642711e-2
+1.703125, 9.35503096739197262423942517026e-2
+1.7109375, 9.23109880289671692896353929825e-2
+1.71875, 9.10825250760733619125242622929e-2
+1.7265625, 8.98649252656317980248107604369e-2
+1.734375, 8.86581910129419280134516806045e-2
+1.7421875, 8.74623227248282428886786528996e-2
+1.75, 8.6277318826511508123032629127e-2
+1.7578125, 8.5103175788718987661607675968e-2
+1.765625, 8.39398881550205742745426187726e-2
+1.7734375, 8.27874485693778234355634874125e-2
+1.78125, 8.16458478038929741309473635317e-2
+1.7890625, 8.05150747867451313139008605519e-2
+1.796875, 7.93951166303008810007929709849e-2
+1.8046875, 7.8285958659386710137338135983e-2
+1.8125, 7.71875844397107092227930950506e-2
+1.8203125, 7.60999758064211461454274743077e-2
+1.828125, 7.50231128927896146287838615296e-2
+1.8359375, 7.39569741590065799904004189494e-2
+1.84375, 7.29015364210772684466786681165e-2
+1.8515625, 7.1856774879805973832596859345e-2
+1.859375, 7.08226631498569872154520993245e-2
+1.8671875, 6.97991732888804903396147021085e-2
+1.875, 6.87862758266918930157383417828e-2
+1.8828125, 6.77839397944932373339145721951e-2
+1.890625, 6.67921327541254378064899491149e-2
+1.8984375, 6.58108208273402761030438211608e-2
+1.90625, 6.48399687250812217975757699202e-2
+1.9140625, 6.38795397767623063764386258326e-2
+1.921875, 6.29294959595344365251722783004e-2
+1.9296875, 6.19897979275286942934588572172e-2
+1.9375, 6.10604050410663360004471155013e-2
+1.9453125, 6.01412753958253685584842279995e-2
+1.953125, 5.92323658519537511330152646697e-2
+1.9609375, 5.83336320631194415916804934726e-2
+1.96875, 5.74450285054876808986011642098e-2
+1.9765625, 5.65665085066160843532421168719e-2
+1.984375, 5.56980242742582862305006120446e-2
+1.9921875, 5.4839526925067063823975007007e-2
+2.0, 5.39909665131880480027201663569e-2
+2.0078125, 5.31522920587353100391014360956e-2
+2.015625, 5.23234515761402985284840131278e-2
+2.0234375, 5.15043921023657855783409750197e-2
+2.03125, 5.06950597249766679739308873917e-2
+2.0390625, 4.98953996100596566101466282519e-2
+2.046875, 4.91053560299840759958565776417e-2
+2.0546875, 4.83248723909961849707171616194e-2
+2.0625, 4.75538912606396198090433740095e-2
+2.0703125, 4.67923543949947515062909139541e-2
+2.078125, 4.60402027657299401378919906224e-2
+2.0859375, 4.52973765869578606359635047802e-2
+2.09375, 4.45638153418902660366933674116e-2
+2.1015625, 4.38394578092847461015341470229e-2
+2.109375, 4.31242420896772311018694814447e-2
+2.1171875, 4.2418105631394182374439277868e-2
+2.125, 4.17209852563386029001239936572e-2
+2.1328125, 4.10328171855441925300841391629e-2
+2.140625, 4.03535370644921634809338368885e-2
+2.1484375, 3.96830799881854222466562921282e-2
+2.15625, 3.90213805259750140332929853937e-2
+2.1640625, 3.83683727461339151189279783878e-2
+2.171875, 3.772399024017344708396773354e-2
+2.1796875, 3.70881661468977745549912850286e-2
+2.1875, 3.64608331761921348713308442855e-2
+2.1953125, 3.58419236325406338308889890592e-2
+2.203125, 3.52313694382696263164134852663e-2
+2.2109375, 3.46291021565128840635220148839e-2
+2.21875, 3.40350530138949350272832042702e-2
+2.2265625, 3.34491529229291396573207743369e-2
+2.234375, 3.28713325041272488265505260118e-2
+2.2421875, 3.23015221078173661022682368411e-2
+2.25, 3.17396518356674134290225263008e-2
+2.2578125, 3.11856515619113740413421601457e-2
+2.265625, 3.06394509542757594739323461033e-2
+2.2734375, 3.01009794946039188225850202531e-2
+2.28125, 2.95701664991759778681304121949e-2
+2.2890625, 2.90469411387223632478522055662e-2
+2.296875, 2.85312324581290324856536819793e-2
+2.3046875, 2.80229693958326943178514148838e-2
+2.3125, 2.75220808029044653219359438039e-2
+2.3203125, 2.70284954618205683193176846352e-2
+2.328125, 2.65421421049188353305017317447e-2
+2.3359375, 2.60629494325399329650202631393e-2
+2.34375, 2.55908461308523809836841681899e-2
+2.3515625, 2.51257608893605853343507939243e-2
+2.359375, 2.46676224180952551936529715313e-2
+2.3671875, 2.42163594644857194073508705299e-2
+2.375, 2.37719008299138011746394329972e-2
+2.3828125, 2.33341753859490508324741752474e-2
+2.390625, 2.29031120902652751324727984527e-2
+2.3984375, 2.24786400022384374350001130184e-2
+2.40625, 2.20606882982261367445070614516e-2
+2.4140625, 2.16491862865290044509767907297e-2
+2.421875, 2.12440634220344860003650207511e-2
+2.4296875, 2.08452493205436004701478097426e-2
+2.4375, 2.04526737727813941544303991346e-2
+2.4453125, 2.00662667580919247484095667019e-2
+2.453125, 1.96859584578187305481380774104e-2
+2.4609375, 1.9311679268371854234243053932e-2
+2.46875, 1.89433598139826032751141434916e-2
+2.4765625, 1.85809309591473387555728897236e-2
+2.484375, 1.82243238207616915024612646502e-2
+2.4921875, 1.78734697799467087320444755929e-2
+2.5, 1.75283004935685360804810157652e-2
+2.5078125, 1.71887479054533387944817369339e-2
+2.515625, 1.68547442572992620529789440341e-2
+2.5234375, 1.6526222099287323862153163076e-2
+2.53125, 1.62031143003932247171616526969e-2
+2.5390625, 1.58853540584021462576337585078e-2
+2.546875, 1.55728749096286964652778439874e-2
+2.5546875, 1.52656107383442415671620720212e-2
+2.5625, 1.49634957859139447252770482337e-2
+2.5703125, 1.46664646596459088212288131967e-2
+2.578125, 1.43744523413548951951535476391e-2
+2.5859375, 1.40873941956431620824039279939e-2
+2.59375, 1.38052259779010357238144584374e-2
+2.6015625, 1.35278838420298937203261586539e-2
+2.609375, 1.32553043478903041766539338378e-2
+2.6171875, 1.29874244684781255489878390723e-2
+2.625, 1.27241815968314308971313793644e-2
+2.6328125, 1.24655135526711764618822187675e-2
+2.640625, 1.22113585887785881648895928449e-2
+2.6484375, 1.19616553971122907828277219372e-2
+2.65625, 1.17163431146682532037308329934e-2
+2.6640625, 1.14753613290856693550068650902e-2
+2.671875, 1.12386500840019381252488289799e-2
+2.6796875, 1.10061498841599469117246315547e-2
+2.6875, 1.07778017002709023395046110399e-2
+2.6953125, 1.05535469736359882446278104031e-2
+2.703125, 1.03333276205301652214130865354e-2
+2.7109375, 1.01170860363514579327058689683e-2
+2.71875, 9.90476509953910600201191162977e-3
+2.7265625, 9.69630817526398167934528721325e-3
+2.734375, 9.49165911889470263015591670917e-3
+2.7421875, 9.29076227924289117152010793619e-3
+2.75, 9.09356250159105210512944562976e-3
+2.7578125, 8.90000513050656000635262185327e-3
+2.765625, 8.7100360124452634571901515546e-3
+2.7734375, 8.52360149814822829324238925573e-3
+2.78125, 8.34064844483515450631170745123e-3
+2.7890625, 8.16112421819801204086677603585e-3
+2.796875, 7.98497669419844938064539560464e-3
+2.8046875, 7.81215426067253557789197002151e-3
+2.8125, 7.6426058187464012692081905811e-3
+2.8203125, 7.47628078406634728616223972325e-3
+2.828125, 7.31312908784699074337192422736e-3
+2.8359375, 7.15310117774101800973599300399e-3
+2.84375, 6.9961480185341117780561413377e-3
+2.8515625, 6.84222109266861558289531363161e-3
+2.859375, 6.69127240059949361475619110187e-3
+2.8671875, 6.54325446098613657928288971282e-3
+2.875, 6.39812031072355569205109443675e-3
+2.8828125, 6.25582350481649672156745923262e-3
+2.890625, 6.11631811609999433435879565465e-3
+2.8984375, 5.97955873480987389553916341573e-3
+2.90625, 5.84550046800669337504873778301e-3
+2.9140625, 5.71409893885660214289293654821e-3
+2.921875, 5.58531028577257624515934532282e-3
+2.9296875, 5.4590911614194712752691374242e-3
+2.9375, 5.33539873158631423065037877059e-3
+2.9453125, 5.2141906739292348125063969512e-3
+2.953125, 5.09542517658841452415591460096e-3
+2.9609375, 4.97906093668240868994219447404e-3
+2.96875, 4.86505715868317219016016817183e-3
+2.9765625, 4.75337355267509432584125734266e-3
+2.984375, 4.64397033250132182834632051309e-3
+2.9921875, 4.53680821380062165008207299777e-3
+3.0, 4.43184841193800685154358059345e-3
+3.0078125, 4.32905263983231967327846776869e-3
+3.015625, 4.22838310568393578594803955797e-3
+3.0234375, 4.12980251060572278378722696095e-3
+3.03125, 4.03327404616035426246291342973e-3
+3.0390625, 3.9387613918070483372696015505e-3
+3.046875, 3.84622871226076624708489066679e-3
+3.0546875, 3.75564065476687278845388851304e-3
+3.0625, 3.66696234629422576710293515814e-3
+3.0703125, 3.58015939064962647520914684685e-3
+3.078125, 3.49519786551652743555936397817e-3
+3.0859375, 3.41204431942085733156959194983e-3
+3.09375, 3.33066576862678619780505571615e-3
+3.1015625, 3.25102969396521661148308200074e-3
+3.109375, 3.17310403759774883332755498827e-3
+3.1171875, 3.09685719971882962746588604119e-3
+3.125, 3.02225803519875587571895230747e-3
+3.1328125, 2.94927585017016512203454712318e-3
+3.140625, 2.87788039856060586785213434431e-3
+3.1484375, 2.80804187857374081824333015924e-3
+3.15625, 2.73973092912169638060926091935e-3
+3.1640625, 2.67291862621103157086537166323e-3
+3.171875, 2.60757647928475911420483779696e-3
+3.1796875, 2.54367642752281096596223456146e-3
+3.1875, 2.48119083610329974951364380869e-3
+3.1953125, 2.42009249242688673871286781234e-3
+3.203125, 2.36035460230652602768768710029e-3
+3.2109375, 2.30195078612481345596034646905e-3
+3.21875, 2.24485507496112771630828396862e-3
+3.2265625, 2.18904190669070989047802360813e-3
+3.234375, 2.13448612205778645717646995319e-3
+3.2421875, 2.08116296072479962049316633681e-3
+3.25, 2.02904805729976763729112390679e-3
+3.2578125, 1.97811743734375670081173388754e-3
+3.265625, 1.92834751336040488587232838193e-3
+3.2734375, 1.87971508076939769912674031775e-3
+3.28125, 1.83219731386575392587616456149e-3
+3.2890625, 1.78577176176673974225997204466e-3
+3.296875, 1.74041634434818848715924255498e-3
+3.3046875, 1.6961093481719630800818683348e-3
+3.3125, 1.65282942240625784737826561601e-3
+3.3203125, 1.61055557474039649651348501788e-3
+3.328125, 1.5692671672957431733914191136e-3
+3.3359375, 1.52894391253430396693366734062e-3
+3.34375, 1.48956586916655690375387440755e-3
+3.3515625, 1.45111343806000941878681937095e-3
+3.359375, 1.41356735814994350953741270801e-3
+3.3671875, 1.37690870235377029607799548819e-3
+3.375, 1.34111887349037752938018694375e-3
+3.3828125, 1.30617960020581572982943928376e-3
+3.390625, 1.27207293290663110812318361868e-3
+3.3984375, 1.23878123970211623396641440097e-3
+3.40625, 1.20628720235671258530946357319e-3
+3.4140625, 1.17457381225376264307336346303e-3
+3.421875, 1.14362436637177310363053587682e-3
+3.4296875, 1.11342246327431507351090751569e-3
+3.4375, 1.08395199911465179715709788326e-3
+3.4453125, 1.0551971636561495578476615964e-3
+3.453125, 1.02714243630949289246134141435e-3
+3.4609375, 9.99772582187691180417952097523e-4
+3.46875, 9.7307264817983001329323400161e-4
+3.4765625, 9.47027959044487531203727028745e-4
+3.484375, 9.21624113523703131586322718751e-4
+3.4921875, 8.96846980478353621511039757078e-4
+3.5, 8.7268269504576000179023728196e-4
+3.5078125, 8.49117654820316645086289987204e-4
+3.515625, 8.26138516057903665761755148737e-4
+3.5234375, 8.0373218990481278074394648833e-4
+3.53125, 7.8188583865188693217391140852e-4
+3.5390625, 7.60586872014544387672123202881e-4
+3.546875, 7.39822943439328955891649990452e-4
+3.5546875, 7.19581946437599356482116640698e-4
+3.5625, 6.99852010946942667126722628603e-4
+3.5703125, 6.80621499720869138054815055839e-4
+3.578125, 6.61879004747318518139873148029e-4
+3.5859375, 6.43613343696481377792352632373e-4
+3.59375, 6.25813556398412743441430416402e-4
+3.6015625, 6.08468901350889677238076188463e-4
+3.609375, 5.91568852257939244157264530736e-4
+3.6171875, 5.75103094599438607075528346349e-4
+3.625, 5.59061522232164778498605865639e-4
+3.6328125, 5.43434234022647834969334220774e-4
+3.640625, 5.28211530512158166072733699783e-4
+3.6484375, 5.13383910614135583374113457593e-4
+3.65625, 4.98942068344345854312155715808e-4
+3.6640625, 4.84876889584028450500120599144e-4
+3.671875, 4.71179448876278007293244655632e-4
+3.6796875, 4.57841006255881179846780672239e-4
+3.6875, 4.44853004112810247971742687344e-4
+3.6953125, 4.3220706408955496542420666354e-4
+3.703125, 4.19894984012454766151999928332e-4
+3.7109375, 4.0790873485717452757381309001e-4
+3.71875, 3.96240457748448646083031194073e-4
+3.7265625, 3.8488246099420019936132378712e-4
+3.734375, 3.73827217154124450278916607918e-4
+3.7421875, 3.63067360142808884494980168585e-4
+3.75, 3.52595682367445364527529524602e-4
+3.7578125, 3.42405131900173823049187512278e-4
+3.765625, 3.32488809685081203338722299778e-4
+3.7734375, 3.22839966779864080885296763274e-4
+3.78125, 3.13452001632148562668112087688e-4
+3.7890625, 3.04318457390446655049237970476e-4
+3.796875, 2.95433019249714312824849473452e-4
+3.8046875, 2.8678951183146282596189571869e-4
+3.8125, 2.78381896598362061971267010294e-4
+3.8203125, 2.7020426930326135569504227108e-4
+3.828125, 2.62250857472541519373470851129e-4
+3.8359375, 2.54516017923699528971250886867e-4
+3.84375, 2.4699423431705592255811960179e-4
+3.8515625, 2.3968011474146381764902384757e-4
+3.859375, 2.32568389333887711330987506507e-4
+3.8671875, 2.25653907932709864183447646328e-4
+3.875, 2.18931637764612080817673598371e-4
+3.8828125, 2.12396661164871080641090839771e-4
+3.890625, 2.06044173330896396462387646776e-4
+3.8984375, 1.99869480108830840013356290439e-4
+3.90625, 1.93867995813025026550624842249e-4
+3.9140625, 1.88035241078189249553958467194e-4
+3.921875, 1.82366840744018135263919976024e-4
+3.9296875, 1.76858521772075979478610129117e-4
+3.9375, 1.71506111194723469711907526781e-4
+3.9453125, 1.66305534095859618540390925583e-4
+3.953125, 1.612528116232461727550984265e-4
+3.9609375, 1.56344059032175511799831395732e-4
+3.96875, 1.51575483760237101926701116268e-4
+3.9765625, 1.46943383532931923537508598544e-4
+3.984375, 1.42444144499878932314423933941e-4
+3.9921875, 1.38074239401352543989869629037e-4
+4.0, 1.33830225764885341988347495891e-4
+4.0078125, 1.29708744131665690730909536438e-4
+4.015625, 1.25706516312455689182198446433e-4
+4.0234375, 1.21820343672750913317417724867e-4
+4.03125, 1.18047105446899666693263313689e-4
+4.0390625, 1.14383757080895979351029577456e-4
+4.046875, 1.10827328603557361020349251618e-4
+4.0546875, 1.07374923025795319220109475438e-4
+4.0625, 1.04023714767683890596799528312e-4
+4.0703125, 1.0077094811302889896641130775e-4
+4.078125, 9.76139356911383403418657791457e-5
+4.0859375, 9.45500569854921980854046978439e-5
+4.09375, 9.15767568690081046222558851721e-5
+4.1015625, 8.86915441655975843343657767047e-5
+4.109375, 8.58919902377061298178899772745e-5
+4.1171875, 8.31757275995290751848710182371e-5
+4.125, 8.05404485555941301221279274183e-5
+4.1328125, 7.79839038644005216492860729612e-5
+4.140625, 7.55039014268039516617269523268e-5
+4.1484375, 7.30983049988360121815168846029e-5
+4.15625, 7.07650329286463016099624475809e-5
+4.1640625, 6.8502056917255249082086759651e-5
+4.171875, 6.6307400802805575233012102923e-5
+4.1796875, 6.41791393680003913314824915555e-5
+4.1875, 6.21153971704161599157957768591e-5
+4.1953125, 6.01143473953791039492991225678e-5
+4.203125, 5.81742107310941533322490770668e-5
+4.2109375, 5.62932542657161526699603523784e-5
+4.21875, 5.44697904060538178801379900798e-5
+4.2265625, 5.270217581759781697292173786e-5
+4.234375, 5.09888103855653576754359014784e-5
+4.2421875, 4.93281361966547870912826270319e-5
+4.25, 4.77186365412049419505342299652e-5
+4.2578125, 4.61588349354553279572949280929e-5
+4.265625, 4.46472941636046490939892496106e-5
+4.2734375, 4.31826153393667483829720482371e-5
+4.28125, 4.17634369867246565005128166861e-5
+4.2890625, 4.03884341395851698244689647006e-5
+4.296875, 3.90563174600381910890903819769e-5
+4.3046875, 3.77658323749269600078327890996e-5
+4.3125, 3.65157582304372742726677004404e-5
+4.3203125, 3.5304907464415849586445210158e-5
+4.328125, 3.41321247961300872490260993776e-5
+4.3359375, 3.29962864331837057890201263378e-5
+4.34375, 3.18962992953049457769834225027e-5
+4.3515625, 3.08311002547263709136272944532e-5
+4.359375, 2.97996553928776604733508199222e-5
+4.3671875, 2.88009592731152149888903624192e-5
+4.375, 2.78340342292148755506724644297e-5
+4.3828125, 2.68979296693565842016943859471e-5
+4.390625, 2.59917213953323856456786461353e-5
+4.3984375, 2.51145109367117859357318622666e-5
+4.40625, 2.4265424899701139127792506806e-5
+4.4140625, 2.34436143304364252943747348226e-5
+4.421875, 2.26482540924515100971181820734e-5
+4.4296875, 2.18785422580667346794559515368e-5
+4.4375, 2.11336995134454724011044838993e-5
+4.4453125, 2.04129685770691034009536465252e-5
+4.453125, 1.97156136313836967195985207946e-5
+4.4609375, 1.90409197673745503802038669926e-5
+4.46875, 1.83881924418276201265752668961e-5
+4.4765625, 1.77567569470397652263578569997e-5
+4.484375, 1.71459578927426527067202673099e-5
+4.4921875, 1.65551587000080875058214121275e-5
+4.5, 1.59837411069054732656755500466e-5
+4.5078125, 1.54311046856850548935174475522e-5
+4.515625, 1.48966663712635476742712383293e-5
+4.5234375, 1.43798600007917167822703339891e-5
+4.53125, 1.38801358640864337322784273498e-5
+4.5390625, 1.33969602647127009038279510685e-5
+4.546875, 1.29298150915041001031007835707e-5
+4.5546875, 1.24781974003130845847542409607e-5
+4.5625, 1.2041619005785494490586495703e-5
+4.5703125, 1.1619606082956631776755634875e-5
+4.578125, 1.12116987784691809551552376454e-5
+4.5859375, 1.08174508312162049800138610975e-5
+4.59375, 1.04364292022153800331398030583e-5
+4.6015625, 1.00682137135235575176303044892e-5
+4.609375, 9.71239669600365502833184797943e-6
+4.6171875, 9.36858264575877924580102068477e-6
+4.625, 9.03638788905137146582529544384e-6
+4.6328125, 8.71544025552803974352337978053e-6
+4.640625, 8.40537875957359936143568084391e-6
+4.6484375, 8.1058532896206845326198511454e-6
+4.65625, 7.81652430524411797539715923092e-6
+4.6640625, 7.53706254187203034292459153963e-6
+4.671875, 7.26714872294850759815831071419e-6
+4.6796875, 7.00647327938531047511606292802e-6
+4.6875, 6.75473607614295538389493227507e-6
+4.6953125, 6.51164614578416976312197646341e-6
+4.703125, 6.27692142884543626213334722142e-6
+4.7109375, 6.0502885208750185872311778469e-6
+4.71875, 5.83148242598851675027081369429e-6
+4.7265625, 5.62024631679563022844124820797e-6
+4.734375, 5.41633130055441363179498427437e-6
+4.7421875, 5.21949619141189036483780091314e-6
+4.75, 5.02950728859244497897488581824e-6
+4.7578125, 4.84613816039794399529110887907e-6
+4.765625, 4.66916943388603751538505820616e-6
+4.7734375, 4.49838859009556954616375088472e-6
+4.78125, 4.33358976469047328717011645214e-6
+4.7890625, 4.17457355389594833997626392262e-6
+4.796875, 4.02114682560310960068356592598e-6
+4.8046875, 3.8731225355206622184694435418e-6
+4.8125, 3.73031954825449320201612996051e-6
+4.8203125, 3.59256246319837781409714356573e-6
+4.828125, 3.45968144512127762026617837983e-6
+4.8359375, 3.3315120593389567825127739581e-6
+4.84375, 3.20789511135986376849579482961e-6
+4.8515625, 3.08867649089741695989740082837e-6
+4.859375, 2.97370702014299458993598125678e-6
+4.8671875, 2.86284230619606194178861178465e-6
+4.875, 2.75594259754997173879131670871e-6
+4.8828125, 2.6528726445340471158120038395e-6
+4.890625, 2.55350156361460046023277580278e-6
+4.8984375, 2.45770270545955575004693993083e-6
+4.90625, 2.36535352667332681290283433679e-6
+4.9140625, 2.27633546511055921779035994427e-6
+4.921875, 2.19053381867926934113345958567e-6
+4.9296875, 2.10783762754581058773041060487e-6
+4.9375, 2.02813955965596387578868626841e-6
+4.9453125, 1.9513357994882874102337022262e-6
+4.953125, 1.87732593995766957941086898137e-6
+4.9609375, 1.8060128773888086404113063493e-6
+4.96875, 1.73730270948109384339787857748e-6
+4.9765625, 1.67110463618808493348635492564e-6
+4.984375, 1.60733086343648071951840085719e-6
+4.9921875, 1.54589650961113278305235951962e-6
+5.0, 1.48671951473429759919863234918e-6
+5.0078125, 1.42972055226892954462537228031e-6
+5.015625, 1.37482294347739867771997623457e-6
+5.0234375, 1.32195257426857100105827005509e-6
+5.03125, 1.27103781446771537505401095943e-6
+5.0390625, 1.22200943944520057001726292022e-6
+5.046875, 1.17480055404141835549880954607e-6
+5.0546875, 1.12934651872681427152863318336e-6
+5.0625, 1.08558487793732705365871335943e-6
+5.0703125, 1.04345529052693084636187259899e-6
+5.078125, 1.00289946228034159695674910553e-6
+5.0859375, 9.63861080430290639934078787596e-7
+5.09375, 9.26285750125084729547386004978e-7
+5.1015625, 8.90120932793462931709385371447e-7
+5.109375, 8.55315886355027123859128862628e-7
+5.1171875, 8.21821607225764656784557420883e-7
+5.125, 7.89590774069399292304307011809e-7
+5.1328125, 7.58577693246500135457226102597e-7
+5.140625, 7.2873824591444822264550472196e-7
+5.1484375, 7.00029836732507003946337635176e-7
+5.15625, 6.72411344127366466857663648875e-7
+5.1640625, 6.45843072075631390487016844861e-7
+5.171875, 6.20286703360802495861279398563e-7
+5.1796875, 5.95705254263355373412178855252e-7
+5.1875, 5.72063030643556327850487966405e-7
+5.1953125, 5.49325585377666989691971152147e-7
+5.203125, 5.27459677109181008546835654762e-7
+5.2109375, 5.06433230277706673400801729791e-7
+5.21875, 4.86215296389059207306715601263e-7
+5.2265625, 4.66776016491056066099594561222e-7
+5.234375, 4.48086584820418140788287610173e-7
+5.2421875, 4.30119213587069628722080199272e-7
+5.25, 4.12847098862999806563452916785e-7
+5.2578125, 3.96244387543701314943034374493e-7
+5.265625, 3.80286145351032156019854373773e-7
+5.2734375, 3.64948325847162715608014258045e-7
+5.28125, 3.50207740430065054486761042949e-7
+5.2890625, 3.36042029281779771096837245814e-7
+5.296875, 3.22429633241456220693403110829e-7
+5.3046875, 3.09349766575905083226721115876e-7
+5.3125, 2.96782390621128501076270410414e-7
+5.3203125, 2.84708188269002553734681382144e-7
+5.328125, 2.73108539273979993110816152834e-7
+5.3359375, 2.61965496355358221692750649363e-7
+5.34375, 2.51261762071318745581013018952e-7
+5.3515625, 2.40980666441590062272264232363e-7
+5.359375, 2.31106145296216433552865195189e-7
+5.3671875, 2.21622719328530528977125487902e-7
+5.375, 2.12515473831028784616376676462e-7
+5.3828125, 2.03770039093434781886550283211e-7
+5.390625, 1.95372571442808286388318148132e-7
+5.3984375, 1.87309734906116068131323625036e-7
+5.40625, 1.79568683476225520716413658303e-7
+5.4140625, 1.72137043962813673559094460517e-7
+5.421875, 1.65002899410202710629064519414e-7
+5.4296875, 1.58154773064638831012931019999e-7
+5.4375, 1.51581612874024467375099584096e-7
+5.4453125, 1.45272776503594771484954643149e-7
+5.453125, 1.39218016851498131642894719569e-7
+5.4609375, 1.33407468048697552141787365692e-7
+5.46875, 1.27831631928055243703749061454e-7
+5.4765625, 1.22481364947896986759140418124e-7
+5.484375, 1.17347865555775973850527492248e-7
+5.4921875, 1.1242266197856814743498045839e-7
+5.5, 1.07697600425432755712922771817e-7
+5.5078125, 1.03164833690563179309844834885e-7
+5.515625, 9.88168101430342598361274072386e-8
+5.5234375, 9.46462630914236083778876466766e-8
+5.53125, 9.06462005112459053209370657423e-8
+5.5390625, 8.68098951235912367354162455338e-8
+5.546875, 8.31308748137012576669186312669e-8
+5.5546875, 7.96029133785506365711616226724e-8
+5.5625, 7.62200215928260219418630851913e-8
+5.5703125, 7.29764385830108827384990482081e-8
+5.578125, 6.98666234995922060285098040263e-8
+5.5859375, 6.68852474777043825178291958955e-8
+5.59375, 6.40271858768168642615741747158e-8
+5.6015625, 6.12875107903555264517187180824e-8
+5.609375, 5.86614838164232911362609176207e-8
+5.6171875, 5.6144549081053656155571962552e-8
+5.625, 5.3732326505691495381600409745e-8
+5.6328125, 5.14206053108490312050876610582e-8
+5.640625, 4.92053377481313986432837942393e-8
+5.6484375, 4.70826330530658907854526185433e-8
+5.65625, 4.50487516114019627714551438896e-8
+5.6640625, 4.31000993317755382177883147317e-8
+5.671875, 4.12332222178512669928128785843e-8
+5.6796875, 3.94448011332702824731900270637e-8
+5.6875, 3.77316467529388528380527553367e-8
+5.6953125, 3.60906946943952645341522693916e-8
+5.703125, 3.4519000823188463769734642905e-8
+5.7109375, 3.30137367263925578321099527894e-8
+5.71875, 3.15721853485663833403872553185e-8
+5.7265625, 3.01917367846471215420872728516e-8
+5.734375, 2.88698842244415169102514147991e-8
+5.7421875, 2.76042200435477672689678607861e-8
+5.75, 2.63924320357057313802261546013e-8
+5.7578125, 2.52322997817328705467924703548e-8
+5.765625, 2.41216911503584288159343736223e-8
+5.7734375, 2.30585589264188836433722607667e-8
+5.78125, 2.20409375620237846126901383751e-8
+5.7890625, 2.106694004644285863696820032e-8
+5.796875, 2.01347548906028100853162960659e-8
+5.8046875, 1.92426432222156950574244795667e-8
+5.8125, 1.83889359876902096829994354522e-8
+5.8203125, 1.75720312571028095248489511581e-8
+5.828125, 1.67903916286273751927287410849e-8
+5.8359375, 1.60425417289402600492107055957e-8
+5.84375, 1.53270658062320990066707890971e-8
+5.8515625, 1.46426054125688201900477550502e-8
+5.859375, 1.39878571724519787409489391066e-8
+5.8671875, 1.33615706345329171237318890599e-8
+5.875, 1.27625462035364396506036499892e-8
+5.8828125, 1.21896331495477591235259028611e-8
+5.890625, 1.16417276919115169516417630304e-8
+5.8984375, 1.11177711550837792397456595357e-8
+5.90625, 1.06167481938671525283443604827e-8
+5.9140625, 1.01376850855456244846917787162e-8
+5.921875, 9.67964808651949533203264540447e-9
+5.9296875, 9.241741851121901682135731184e-9
+5.9375, 8.8231079103770203466644646543e-9
+5.9453125, 8.42292320853614844592033769115e-9
+5.953125, 8.04039869530155870150255702178e-9
+5.9609375, 7.67477797171939441258791008712e-9
+5.96875, 7.32533598779196475632756998286e-9
+5.9765625, 6.99137778992669350049949997999e-9
+5.984375, 6.6722373164037270810507095417e-9
+5.9921875, 6.36727623910688375300547210793e-9
+6.0, 6.07588284982328504272498960405e-9
+6.0078125, 5.79747098947571331580170369069e-9
+6.015625, 5.53147901870854269447053445049e-9
+6.0234375, 5.27736882830305107343128764573e-9
+6.03125, 5.03462488795109338292392335065e-9
+6.0390625, 4.80275333196755388112728290715e-9
+6.046875, 4.58128108057175006263739778946e-9
+6.0546875, 4.36975499541608330452207551359e-9
+6.0625, 4.16774106808677084544314747702e-9
+6.0703125, 3.9748236403464979961427885126e-9
+6.078125, 3.79060465493234521045091226652e-9
+6.0859375, 3.61470293576441714272512806719e-9
+6.09375, 3.44675349646127418518808933568e-9
+6.1015625, 3.28640687609758411699178509387e-9
+6.109375, 3.13332850117741412874092655971e-9
+6.1171875, 2.98719807283331218425927040616e-9
+6.125, 2.84770897829682089481353837674e-9
+6.1328125, 2.71456772572036516196546114123e-9
+6.140625, 2.58749340146359407462510041525e-9
+6.1484375, 2.46621714898927415884852431053e-9
+6.15625, 2.35048166854476028976631207766e-9
+6.1640625, 2.24004073683494660178964916796e-9
+6.171875, 2.1346587459214558218161724373e-9
+6.1796875, 2.03411026061069389798653785633e-9
+6.1875, 1.93817959362030897585803027822e-9
+6.1953125, 1.84666039783958015451777216219e-9
+6.203125, 1.75935527502435162703128547237e-9
+6.2109375, 1.67607540029135050458729752312e-9
+6.21875, 1.59664016180010973718743381346e-9
+6.2265625, 1.52087681503328815586565560008e-9
+6.234375, 1.44862015110796405775344724982e-9
+6.2421875, 1.37971217857150244711485020714e-9
+6.25, 1.31400181815588378968607665677e-9
+6.2578125, 1.25134460998395795634180776253e-9
+6.265625, 1.19160243273997423160311486961e-9
+6.2734375, 1.134643234334959452094510938e-9
+6.28125, 1.08034077361509345033010736254e-9
+6.2890625, 1.02857437267818527996132890534e-9
+6.296875, 9.79228679379705816423454417812e-10
+6.3046875, 9.32193439625602262295033340457e-10
+6.3125, 8.87363279064327230902507889499e-10
+6.3203125, 8.44637493805178233216473386624e-10
+6.328125, 8.03919849804180773607106870531e-10
+6.3359375, 7.65118390572377530236549460732e-10
+6.34375, 7.28145252874524370779819451821e-10
+6.3515625, 6.92916490098857818143113187184e-10
+6.359375, 6.59351902990804102265231240636e-10
+6.3671875, 6.27374877455262679669569578032e-10
+6.375, 5.96912229143432151255454909299e-10
+6.3828125, 5.67894054551068465976209417792e-10
+6.390625, 5.40253588365588308018817993176e-10
+6.3984375, 5.13927066809568324756407247627e-10
+6.40625, 4.88853596737956624101049735965e-10
+6.4140625, 4.64975030255719592499446554178e-10
+6.421875, 4.4223584463170698790967176132e-10
+6.4296875, 4.20583027293243476377958947247e-10
+6.4375, 3.99965965694356950016840179721e-10
+6.4453125, 3.80336341858644356241699509296e-10
+6.453125, 3.61648031405565286242994999172e-10
+6.4609375, 3.43857006876452763954851873953e-10
+6.46875, 3.2692124518374975065295128836e-10
+6.4765625, 3.10800639013928706682118391975e-10
+6.484375, 2.95456912021239678958944179119e-10
+6.4921875, 2.80853537655869045115353226607e-10
+6.5, 2.66955661476285172150323676068e-10
+6.5078125, 2.53730026801507473497374874187e-10
+6.515625, 2.41144903564770021213675995657e-10
+6.5234375, 2.29170020235568059404156854759e-10
+6.53125, 2.17776498682383271437765936892e-10
+6.5390625, 2.06936791853489016213310019567e-10
+6.546875, 1.96624624158147253700501152389e-10
+6.5546875, 1.86814934435231567802733167961e-10
+6.5625, 1.77483821400852367991533476149e-10
+6.5703125, 1.68608491470927582388876309986e-10
+6.578125, 1.60167208858841292937206800551e-10
+6.5859375, 1.52139247852369940553770050068e-10
+6.59375, 1.44504847177936867686429600432e-10
+6.6015625, 1.37245166363986787444200041296e-10
+6.609375, 1.30342244018857795841380559411e-10
+6.6171875, 1.23778957941975109668159821383e-10
+6.625, 1.17538986990502965752474704735e-10
+6.6328125, 1.11606774626774028123176028584e-10
+6.640625, 1.05967494074874015291944783898e-10
+6.6484375, 1.00607015017697710807096988438e-10
+6.65625, 9.55118717686155248446252389216e-11
+6.6640625, 9.06692328546016447497642683163e-11
+6.671875, 8.60668719502797084734004565783e-11
+6.6796875, 8.16931401048438705263360451392e-11
+6.6875, 7.75369392062159773671543183368e-11
+6.6953125, 7.35876966291070628582147633296e-11
+6.703125, 6.98353410158671167332603122905e-11
+6.7109375, 6.62702791411345438081734944612e-11
+6.71875, 6.28833738133392687895926425812e-11
+6.7265625, 5.96659227680742809327107428062e-11
+6.734375, 5.6609638510232668481671884839e-11
+6.7421875, 5.37066290636138667501796934224e-11
+6.75, 5.09493795884368295208847380993e-11
+6.7578125, 4.83307348288619201515983164639e-11
+6.765625, 4.58438823542201942310971662035e-11
+6.7734375, 4.3482336559180999968349005e-11
+6.78125, 4.12399233895589514659495110623e-11
+6.7890625, 3.91107657618717381708411772851e-11
+6.796875, 3.70892696461132375839137773881e-11
+6.8046875, 3.51701107825042291518434915904e-11
+6.8125, 3.33482220042278144226556171774e-11
+6.8203125, 3.16187811393505019685957274538e-11
+6.828125, 2.99771994662748086370472339251e-11
+6.8359375, 2.84191106981670808292364879091e-11
+6.84375, 2.69403604728568988177178767727e-11
+6.8515625, 2.55369963257136728170421515748e-11
+6.859375, 2.42052581239735843732681852576e-11
+6.8671875, 2.2941568941917519257424671302e-11
+6.875, 2.17425263571896651743792575491e-11
+6.8828125, 2.06048941493985362696293342872e-11
+6.890625, 1.95255943829588060790009505222e-11
+6.8984375, 1.85016998569148951167416652849e-11
+6.90625, 1.75304269052371290567065041339e-11
+6.9140625, 1.66091285317997670833415186216e-11
+6.921875, 1.57352878649385563078060395302e-11
+6.9296875, 1.49065119171449079587337049725e-11
+6.9375, 1.41205256360854788567857233846e-11
+6.9453125, 1.33751662337409973168905773605e-11
+6.953125, 1.2668377781037672913885086503e-11
+6.9609375, 1.19982060558995098492648972137e-11
+6.96875, 1.13627936331812993593719058174e-11
+6.9765625, 1.07603752054509545918268385266e-11
+6.984375, 1.01892731240770914295065152892e-11
+6.9921875, 9.64789315054423489929160286107e-12
+7.0, 9.13472040836459267493370191438e-12
+7.0078125, 8.64831552638280121500805881497e-12
+7.015625, 8.18731096467920077359416207238e-12
+7.0234375, 7.75040751466878663051256008848e-12
+7.03125, 7.33637096536773953981408595347e-12
+7.0390625, 6.94402892815805420398831442389e-12
+7.046875, 6.57226781272392870837741152512e-12
+7.0546875, 6.2200299471618920300571139077e-12
+7.0625, 5.8863108355807472658266041629e-12
+7.0703125, 5.57015654680788692159162231966e-12
+7.078125, 5.27066122810596172767317323005e-12
+7.0859375, 4.98696473807880157881986000847e-12
+7.09375, 4.71825039320842011929369981444e-12
+7.1015625, 4.46374282271638599311426585229e-12
+7.109375, 4.22270592668329646162026688249e-12
+7.1171875, 3.99444093259000614218986573448e-12
+7.125, 3.77828454566408955552479883543e-12
+7.1328125, 3.57360718862517753702771161725e-12
+7.140625, 3.37981132662371365320176863254e-12
+7.1484375, 3.19632987335972024416306332977e-12
+7.15625, 3.02262467455172130121045670685e-12
+7.1640625, 2.85818506510140244973367085954e-12
+7.171875, 2.70252649646724345614448675549e-12
+7.1796875, 2.55518923092056834758847287468e-12
+7.1875, 2.41573709951054123278609576582e-12
+7.1953125, 2.28375632071089795452500284198e-12
+7.203125, 2.15885437686093791973636340437e-12
+7.2109375, 2.04065894564678788345770858991e-12
+7.21875, 1.92881688399645953630306719153e-12
+7.2265625, 1.82299326188401373316026146992e-12
+7.234375, 1.72287044365446365993400567148e-12
+7.2421875, 1.62814721459213442996732168968e-12
+7.25, 1.53853795056127491375877474721e-12
+7.2578125, 1.45377182864900692587818143923e-12
+7.265625, 1.37359207683740600052690061512e-12
+7.2734375, 1.29775526082383691267613101553e-12
+7.28125, 1.22603060619680747770317002197e-12
+7.2890625, 1.15819935425873956022881895741e-12
+7.296875, 1.09405414986736247655094284743e-12
+7.3046875, 1.03339845974407949332301670552e-12
+7.3125, 9.76046019770804190959230504933e-13
+7.3203125, 9.21820309866564519382320545484e-13
+7.328125, 8.70554055101776311870468451844e-13
+7.3359375, 8.22088751771636432194999902132e-13
+7.34375, 7.76274217210714166167264137034e-13
+7.3515625, 7.32968162188657695641456117875e-13
+7.359375, 6.9203578478210472163053589143e-13
+7.3671875, 6.53349384670511417985365884122e-13
+7.375, 6.16787996853805675165232631032e-13
+7.3828125, 5.82237043837637914933106126272e-13
+7.390625, 5.4958800537764980613026824597e-13
+7.3984375, 5.18738104917707672738334097475e-13
+7.40625, 4.8959001189854862731111903003e-13
+7.4140625, 4.62051559152854719110350802866e-13
+7.421875, 4.36035474640490926946904431108e-13
+7.4296875, 4.11459126813600123535956961752e-13
+7.4375, 3.88244282935522023167752193491e-13
+7.4453125, 3.66316879710169942373904833396e-13
+7.453125, 3.45606805609631957630482893953e-13
+7.4609375, 3.26047694317431537881857570723e-13
+7.46875, 3.07576728733153699765097651242e-13
+7.4765625, 2.90134455011079980266908023653e-13
+7.484375, 2.73664606131140033138747250423e-13
+7.4921875, 2.58113934524937726436677991641e-13
+7.5, 2.43432053302900964787598960194e-13
+7.5078125, 2.295712856507903278861132264e-13
+7.515625, 2.16486521984932888724773592688e-13
+7.5234375, 2.04135084475672870481476980001e-13
+7.53125, 1.92476598567696572879589032967e-13
+7.5390625, 1.81472871144139626831412133523e-13
+7.546875, 1.71087774998762515546140295274e-13
+7.5546875, 1.6128713929702592733819506034e-13
+7.5625, 1.52038645722649559336219293858e-13
+7.5703125, 1.43311730021233412810412891016e-13
+7.578125, 1.35077488666794687222420726957e-13
+7.5859375, 1.27308590390659777482097184643e-13
+7.59375, 1.19979192325081771341451100504e-13
+7.6015625, 1.13064860526259939649481292661e-13
+7.609375, 1.06542494653148327320690344347e-13
+7.6171875, 1.00390256589583671872715244863e-13
+7.625, 9.45875028078652106464067817884e-14
+7.6328125, 8.91147202820059814071122179198e-14
+7.640625, 8.39534657684713041991617931741e-14
+7.6484375, 7.90863082813484727769708525893e-14
+7.65625, 7.4496774597574435863390018689e-14
+7.6640625, 7.01692976361065495935134227997e-14
+7.671875, 6.608916756277549900731463535e-14
+7.6796875, 6.22424854800284561797050072708e-14
+7.6875, 5.86161195678728162065855165901e-14
+7.6953125, 5.51976635490839328429734720851e-14
+7.703125, 5.19753973581608563217769967897e-14
+7.7109375, 4.89382498996180763275802487511e-14
+7.71875, 4.60757637870038738002200183596e-14
+7.7265625, 4.33780619595515608641353224633e-14
+7.734375, 4.08358160786124392251856815957e-14
+7.7421875, 3.8440216611001890530722937801e-14
+7.75, 3.61829445111251693441084546338e-14
+7.7578125, 3.40561444182491554133209573318e-14
+7.765625, 3.20523992895619312280105187215e-14
+7.7734375, 3.0164706393724442820879567474e-14
+7.78125, 2.83864545934780249827557956047e-14
+7.7890625, 2.67114028495380876107971131219e-14
+7.796875, 2.51336598814871635770880776968e-14
+7.8046875, 2.36476649246887620099750249495e-14
+7.8125, 2.22481695253855821884048882055e-14
+7.8203125, 2.09302203191297459847129721179e-14
+7.828125, 1.96891427405265387928341604236e-14
+7.8359375, 1.85205256149640799355249658849e-14
+7.84375, 1.74202065855563923432734472197e-14
+7.8515625, 1.63842583309531916216163905816e-14
+7.859375, 1.54089755319727308260261791466e-14
+7.8671875, 1.44908625472002790548952573797e-14
+7.875, 1.36266217597700485448449091256e-14
+7.8828125, 1.28131425595181084787432872291e-14
+7.890625, 1.20474909265632622312558369311e-14
+7.8984375, 1.13268995841469944911480118701e-14
+7.90625, 1.06487586902471517577624827739e-14
+7.9140625, 1.0010607039077511413509622295e-14
+7.921875, 9.4101237451011101052365562862e-15
+7.9296875, 8.84512038362322281781650802223e-15
+7.9375, 8.3135335633940929690896790805e-15
+7.9453125, 7.81341790794560548218919622553e-15
+7.953125, 7.34293942361358359765689381061e-15
+7.9609375, 6.90036923336161928139690150753e-15
+7.96875, 6.48407765662649629162820416406e-15
+7.9765625, 6.09252861645235857122173235215e-15
+7.984375, 5.72427435617369075370167211382e-15
+7.9921875, 5.37795044884864721876942599072e-15
+8.0, 5.0522710835368919185258245109e-15
+8.0078125, 4.74602461336236574668703713703e-15
+8.015625, 4.45806935110363229918451163615e-15
+8.0234375, 4.18732959881490119826697788059e-15
+8.03125, 3.93279189870161497197695261333e-15
+8.0390625, 3.69350149315762987496521988899e-15
+8.046875, 3.4685589825184419709103782004e-15
+8.0546875, 3.25711716969843094693438424314e-15
+8.0625, 3.05837808146144847894133220635e-15
+8.0703125, 2.87159015662491218578874244751e-15
+8.078125, 2.69604559201944483844244856979e-15
+8.0859375, 2.53107783752050791484715624393e-15
+8.09375, 2.37605923193683084199899255971e-15
+8.1015625, 2.23039877198407363219223510963e-15
+8.109375, 2.09354000699235509346888502536e-15
+8.1171875, 1.96495905239424135494261974819e-15
+8.125, 1.84416271541666919074086366643e-15
+8.1328125, 1.73068672675716674097439433157e-15
+8.140625, 1.62409407236266683600472567112e-15
+8.1484375, 1.52397341974916898859363294114e-15
+8.15625, 1.42993763360342919471773834739e-15
+8.1640625, 1.34162237569462860725348926047e-15
+8.171875, 1.25868478439543454324532187748e-15
+8.1796875, 1.18080222936881900034525342444e-15
+8.1875, 1.1076711372201990723148884012e-15
+8.1953125, 1.03900588414462989879375834824e-15
+8.203125, 9.74537751816596860825117988459e-16
+8.2109375, 9.14013942976067534711813793849e-16
+8.21875, 8.57196653359490167723504446082e-16
+8.2265625, 8.03862196808947349748918225852e-16
+8.234375, 7.53800180567244393014842775399e-16
+8.2421875, 7.06812727931856520486588052243e-16
+8.25, 6.62713745596875105103299007857e-16
+8.2578125, 6.2132823315985303384086260862e-16
+8.265625, 5.82491632410200553742012177633e-16
+8.2734375, 5.46049214147950302431991468181e-16
+8.28125, 5.11855500406696260319580708188e-16
+8.2890625, 4.79773720072697838014038886271e-16
+8.296875, 4.49675296003889162829756423288e-16
+8.3046875, 4.2143936185818796791023050359e-16
+8.3125, 3.94952306940383568328379000466e-16
+8.3203125, 3.70107347471307525501404649213e-16
+8.328125, 3.46804122772245954389723446358e-16
+8.3359375, 3.24948314941915941487669722209e-16
+8.34375, 3.04451290683062642775779202936e-16
+8.3515625, 2.85229764011086989852092367822e-16
+8.359375, 2.67205478648322166027681195494e-16
+8.3671875, 2.50304908974863353609714152597e-16
+8.375, 2.34458978470431180015308585191e-16
+8.3828125, 2.19602794641815137991834034465e-16
+8.390625, 2.05675399487188784618714030672e-16
+8.3984375, 1.92619534602193466324293277833e-16
+8.40625, 1.80381420083321892315774237867e-16
+8.4140625, 1.68910546431958285340416956927e-16
+8.421875, 1.58159478707600721170818266527e-16
+8.4296875, 1.48083672221448160132973366424e-16
+8.4375, 1.38641299101816418469951604736e-16
+8.4453125, 1.29793085100883476502963559547e-16
+8.453125, 1.21502156048177714851300830257e-16
+8.4609375, 1.13733893390128998151811347856e-16
+8.46875, 1.06455798287011866591814468677e-16
+8.4765625, 9.96373637688264424008218136866e-17
+8.484375, 9.32499544801844268220957865691e-17
+8.4921875, 8.72666935711878830904002296176e-17
+8.5, 8.16623563166954944229500061213e-17
+8.5078125, 7.64132700703480296766719927137e-17
+8.515625, 7.14972201823507242834853635006e-17
+8.5234375, 6.68933615313598127618369804798e-17
+8.53125, 6.25821353409641198499819274424e-17
+8.5390625, 5.85451909702571974585695180295e-17
+8.546875, 5.47653123859241326694350402814e-17
+8.5546875, 5.12263490401795692051857594816e-17
+8.5625, 4.79131508948461570566998025219e-17
+8.5703125, 4.48115073469089751661092682421e-17
+8.578125, 4.1908089825071953540010405901e-17
+8.5859375, 3.91903978402246926335655401984e-17
+8.59375, 3.66467082853474703404048444101e-17
+8.6015625, 3.42660277922813030713921384274e-17
+8.609375, 3.20380479640090134290409642308e-17
+8.6171875, 2.99531033116704678213072755006e-17
+8.625, 2.80021317355065052423075317431e-17
+8.6328125, 2.61766373983256252808839657233e-17
+8.640625, 2.44686558489474072549463478219e-17
+8.6484375, 2.28707212614272883082672868401e-17
+8.65625, 2.13758356637374540776762721666e-17
+8.6640625, 1.99774400369953220621935611145e-17
+8.671875, 1.86693871733200549925868059994e-17
+8.6796875, 1.74459161869829394256304355233e-17
+8.6875, 1.63016285797221707693966193276e-17
+8.6953125, 1.52314657669381973463662057547e-17
+8.703125, 1.4230687976992689365571681921e-17
+8.7109375, 1.32948544410216746864320041876e-17
+8.71875, 1.24198047955596092152582110991e-17
+8.7265625, 1.16016416248732977818201725625e-17
+8.734375, 1.08367140742388638614129827449e-17
+8.7421875, 1.01216024694766888631426539588e-17
+8.75, 9.45310388190285177802296249465e-18
+8.7578125, 8.82821858147473523813832411158e-18
+8.765625, 8.24413732431599604503362876513e-18
+8.7734375, 7.69822942401417476398466275182e-18
+8.78125, 7.18803155910430390716769404086e-18
+8.7890625, 6.71123727199478518347399949683e-18
+8.796875, 6.2656871172677503777066797532e-18
+8.8046875, 5.84935941980472795158743058248e-18
+8.8125, 5.46036160555879388687669891108e-18
+8.8203125, 5.09692207002506140771069798478e-18
+8.828125, 4.75738255156044471418609162763e-18
+8.8359375, 4.44019097867874233080035074573e-18
+8.84375, 4.14389476230541817208929327238e-18
+8.8515625, 3.86713450572479560857996558611e-18
+8.859375, 3.60863810659711777428646681848e-18
+8.8671875, 3.3672152269700972178263818554e-18
+8.875, 3.14175210866486632819290055372e-18
+8.8828125, 2.93120671278500806155564298097e-18
+8.890625, 2.73460416338464862412027121429e-18
+8.8984375, 2.55103247654219439882095555932e-18
+8.90625, 2.3796385572246855032563892714e-18
+8.9140625, 2.2196244473981507797428876213e-18
+8.921875, 2.07024380984577302085516742401e-18
+8.9296875, 1.93079863310186827069200442217e-18
+8.9375, 1.80063614379919171161490565798e-18
+8.9453125, 1.67914591356324296432746332283e-18
+8.953125, 1.56575714837320061505323533362e-18
+8.9609375, 1.45993614904783240263055839459e-18
+8.96875, 1.36118393220899499761322063779e-18
+8.9765625, 1.26903400172778497437863850634e-18
+8.984375, 1.18305026127150696496915542822e-18
+8.9921875, 1.1028250591457186023519666963e-18
+9.0, 1.02797735716689140436401731321e-18
+9.0078125, 9.5815101580976085056710381021e-19
+9.015625, 8.93013188351175458930593665494e-19
+9.0234375, 8.32252817181027751446320533452e-19
+9.03125, 7.75579225872386384518108771008e-19
+9.0390625, 7.22720800998874050867365417815e-19
+9.046875, 6.73423758059180349706147697274e-19
+9.0546875, 6.27450986217803146719246882486e-19
+9.0625, 5.84580966899031817375006753514e-19
+9.0703125, 5.4460676157909721913005622965e-19
+9.078125, 5.0733506441051713115542295761e-19
+9.0859375, 4.72585315584092270056648229793e-19
+9.09375, 4.40188871588820147619291778109e-19
+9.1015625, 4.09988228769192435601390422553e-19
+9.109375, 3.81836296803867387348107926491e-19
+9.1171875, 3.55595718940447711728203614865e-19
+9.125, 3.31138236018879051927521833406e-19
+9.1328125, 3.08344091501598374644622846348e-19
+9.140625, 2.87101474902742027662356194983e-19
+9.1484375, 2.67306001172163017343267599469e-19
+9.15625, 2.48860223743357936107327643111e-19
+9.1640625, 2.31673179098278796049508680848e-19
+9.171875, 2.15659960836980058119569065153e-19
+9.1796875, 2.00741321366668190353351742503e-19
+9.1875, 1.86843299443489514783570405291e-19
+9.1953125, 1.73896871911790756752266209059e-19
+9.203125, 1.61837628090065709864225900811e-19
+9.2109375, 1.50605465350783856987511236805e-19
+9.21875, 1.40144304533180284767298219292e-19
+9.2265625, 1.30401823914244501219861226907e-19
+9.234375, 1.21329210543929996925263099526e-19
+9.2421875, 1.12880927826346576195096124573e-19
+9.25, 1.05014498299703689733685175948e-19
+9.2578125, 9.76903006343365404301161312163e-20
+9.265625, 9.08713799305412900934959577475e-20
+9.2734375, 8.45232704564283627301544537843e-20
+9.28125, 7.86138300208151172126873066291e-20
+9.2890625, 7.31130852275478791225038093436e-20
+9.296875, 6.79930869057815163536869412392e-20
+9.3046875, 6.32277750558524862124049498089e-20
+9.3125, 5.87928526926464518925470262245e-20
+9.3203125, 5.46656680079605805414366963725e-20
+9.328125, 5.08251043104591367348173846679e-20
+9.3359375, 4.72514772365744899305945290608e-20
+9.34375, 4.39264387582601597706073057173e-20
+9.3515625, 4.08328875439950806916326476763e-20
+9.359375, 3.79548852579975222211848129599e-20
+9.3671875, 3.52775784093537989443806354481e-20
+9.375, 3.278712538781436716680955686e-20
+9.3828125, 3.04706283464644825728154644492e-20
+9.390625, 2.83160696134380816427699956612e-20
+9.3984375, 2.63122523354055913209008666111e-20
+9.40625, 2.44487450748168269540305685334e-20
+9.4140625, 2.27158301009014379428583138199e-20
+9.421875, 2.11044551312988336499339715595e-20
+9.4296875, 1.9606188296979708797935211632e-20
+9.4375, 1.82131761179002297447569709799e-20
+9.4453125, 1.69181042906614588632776068439e-20
+9.453125, 1.57141611023905362859660984959e-20
+9.4609375, 1.45950032971726357258577132727e-20
+9.46875, 1.35547242326963977891944240901e-20
+9.4765625, 1.25878241753797727602815457632e-20
+9.484375, 1.16891825921642479218746339411e-20
+9.4921875, 1.08540323064466739123991644511e-20
+9.5, 1.00779353943000091508660047388e-20
+9.5078125, 9.35676070525539359244302888726e-21
+9.515625, 8.68666289951377637044188369428e-21
+9.5234375, 8.06406290055934770892575256246e-21
+9.53125, 7.48562966879065187796439043489e-21
+9.5390625, 6.94826320799789951008000233365e-21
+9.546875, 6.44907872232421759950049190809e-21
+9.5546875, 5.98539184678020065287716875297e-21
+9.5625, 5.55470487945935453150527085579e-21
+9.5703125, 5.15469394834952552818620816888e-21
+9.578125, 4.78319705007341834943519028103e-21
+9.5859375, 4.43820290203972145708378684431e-21
+9.59375, 4.11784055336381022910401831732e-21
+9.6015625, 3.82036970354093979814076057067e-21
+9.609375, 3.54417168124161474099480550155e-21
+9.6171875, 3.28774103876373811910231514473e-21
+9.625, 3.04967772063353034194179272257e-21
+9.6328125, 2.82867976761048907823079927421e-21
+9.640625, 2.62353651993339982188982988326e-21
+9.6484375, 2.43312228605636597152906322184e-21
+9.65625, 2.25639044537701940155543606194e-21
+9.6640625, 2.09236795556380549647185825575e-21
+9.671875, 1.94015023705515882945508263096e-21
+9.6796875, 1.79889640913952734915103867708e-21
+9.6875, 1.66782485374001526826024935142e-21
+9.6953125, 1.54620908462880649226851663774e-21
+9.703125, 1.43337390129190185458594476958e-21
+9.7109375, 1.32869180806097907259169174441e-21
+9.71875, 1.23157968043284277928465437788e-21
+9.7265625, 1.14149566171403420712780912066e-21
+9.734375, 1.05793627426438602950954901276e-21
+9.7421875, 9.80433730673941221619365522335e-22
+9.75, 9.08553431197666406373285990412e-22
+9.7578125, 8.41891634696419765589810327526e-22
+9.765625, 7.8007329119502040247909219235e-22
+9.7734375, 7.2275002497306782500185428224e-22
+9.78125, 6.69598257855171355231052626888e-22
+9.7890625, 6.20317463068017570897072881978e-22
+9.796875, 5.74628540685546393482782418263e-22
+9.8046875, 5.32272306293524216148537914765e-22
+9.8125, 4.93008085073894459653382406978e-22
+9.8203125, 4.56612404040161966019452747906e-22
+9.828125, 4.2287777565026052655062630267e-22
+9.8359375, 3.91611566485229851509268507838e-22
+9.84375, 3.6263494511280177158186972132e-22
+9.8515625, 3.35781903656732846872268759619e-22
+9.859375, 3.10898347967353718202573308003e-22
+9.8671875, 2.87841251638139541085619790075e-22
+9.875, 2.66477869438827174924452785264e-22
+9.8828125, 2.46685006039289027335698540632e-22
+9.890625, 2.28348336181492424355457292381e-22
+9.8984375, 2.11361772720801951842462791377e-22
+9.90625, 1.95626879203904565105244915639e-22
+9.9140625, 1.81052323879952777490888291854e-22
+9.921875, 1.67553372255250030733848233977e-22
+9.9296875, 1.55051415500990434139297235069e-22
+9.9375, 1.43473532209189181797557758409e-22
+9.9453125, 1.32752081164912117865837038229e-22
+9.953125, 1.22824322964084607992765087715e-22
+9.9609375, 1.1363206845632602600500220835e-22
+9.96875, 1.05121352132158866502692925396e-22
+9.9765625, 9.72421287042734503838461476295e-23
+9.984375, 8.99479912539369640508171716769e-23
+9.9921875, 8.31959094267226163937931266971e-23
+10.0, 7.69459862670641878370648142038e-23
+10.0078125, 7.11612323792387364595118511488e-23
+10.015625, 6.58073561937361269782833017786e-23
+10.0234375, 6.0852569203045942881321233366e-23
+10.03125, 5.62674051101071555849692265929e-23
+10.0390625, 5.20245519064219507143537242908e-23
+10.046875, 4.80986959655039065558281191942e-23
+10.0546875, 4.44663773012588728979796043327e-23
+10.0625, 4.11058552004076879213043218051e-23
+10.0703125, 3.79969834934548101468849799725e-23
+10.078125, 3.51210947802672266782544832556e-23
+10.0859375, 3.24608929743158554388949686495e-23
+10.09375, 3.00003535742916100740217247283e-23
+10.1015625, 2.77246311133683384002812295986e-23
+10.109375, 2.56199732750575521088676824925e-23
+10.1171875, 2.36736412005834164453107581756e-23
+10.125, 2.18738355461856114582465976589e-23
+10.1328125, 2.02096278799046252978258431193e-23
+10.140625, 1.86708970363793148065093216673e-23
+10.1484375, 1.72482700751397785777205173796e-23
+10.15625, 1.59330675129491634454725676348e-23
+10.1640625, 1.4717252524065902592591991602e-23
+10.171875, 1.35933838239841408406797344447e-23
+10.1796875, 1.2554571972377557973256268387e-23
+10.1875, 1.15944388497255673163579898831e-23
+10.1953125, 1.07070800795388722372312538353e-23
+10.203125, 9.88703018431483536171630538514e-24
+10.2109375, 9.12923027842703403034076527006e-24
+10.21875, 8.42899811516689533881169093911e-24
+10.2265625, 7.78200031818214845473927055215e-24
+10.234375, 7.1842266396656609824636121018e-24
+10.2421875, 6.63196609890298029448755018485e-24
+10.25, 6.12178486524712076252918113985e-24
+10.2578125, 5.6505057593102638675696786696e-24
+10.265625, 5.21518925519568862906619514954e-24
+10.2734375, 4.81311587498748141247710142714e-24
+10.28125, 4.4417698745151247240151903136e-24
+10.2890625, 4.09882412665657923409658294398e-24
+10.296875, 3.78212611517571563315761532041e-24
+10.3046875, 3.4896849583440020489585802688e-24
+10.3125, 3.2196593874057866769061482062e-24
+10.3203125, 2.97034661034258224644840644964e-24
+10.328125, 2.74017199640353708632611704816e-24
+10.3359375, 2.52767952152382369545014747584e-24
+10.34375, 2.33152291907516419940133944465e-24
+10.3515625, 2.15045748440656405603900048354e-24
+10.359375, 1.98333248536032949556172161032e-24
+10.3671875, 1.82908413440886891810911675571e-24
+10.375, 1.68672908127047691479635622252e-24
+10.3828125, 1.55535838784480706125949017847e-24
+10.390625, 1.43413195007736472309273810368e-24
+10.3984375, 1.32227333393226039094859102877e-24
+10.40625, 1.21906499503776112235562412846e-24
+10.4140625, 1.12384385378297671404971750216e-24
+10.421875, 1.03599719969851165302791361729e-24
+10.4296875, 9.54958900860440147870154107806e-25
+10.4375, 8.80205895826058638186239070079e-25
+10.4453125, 8.11254947251335118152037387656e-25
+10.453125, 7.47659637862913991814449057892e-25
+10.4609375, 6.89007590870413280131884946399e-25
+10.46875, 6.34917898215434541201688611552e-25
+10.4765625, 5.85038741269505691328547918012e-25
+10.484375, 5.3904518972088681121244049609e-25
+10.4921875, 4.96637165436101851120889406068e-25
+10.5, 4.57537559052080493056792321906e-25
+10.5078125, 4.21490487954353905059767933156e-25
+10.515625, 3.88259685130908539974169808171e-25
+10.5234375, 3.57627009164766617844112203364e-25
+10.53125, 3.29391066345499193274862713143e-25
+10.5390625, 3.03365936544745530678180589563e-25
+10.546875, 2.79379995117179624267170376535e-25
+10.5546875, 2.57274823659731138728787818981e-25
+10.5625, 2.36904202991484078572653308593e-25
+10.5703125, 2.18133182207560239381437220082e-25
+10.578125, 2.00837218115248314432722624588e-25
+10.5859375, 1.84901379782265275286010575451e-25
+10.59375, 1.70219613317749834640567111269e-25
+10.6015625, 1.56694062368630814425077094979e-25
+10.609375, 1.44234440149467590817857560627e-25
+10.6171875, 1.32757449134657517853480401112e-25
+10.625, 1.22186244829839537258774450476e-25
+10.6328125, 1.1244994030605827213826165002e-25
+10.640625, 1.03483148427333170034925613433e-25
+10.6484375, 9.52255589311358148584237761522e-26
+10.65625, 8.76215477332451178645568722191e-26
+10.6640625, 8.06198160247584886298336311988e-26
+10.671875, 7.41730569108319045799410723815e-26
+10.6796875, 6.82376475090648998726262304515e-26
+10.6875, 6.27733645813228880463674625861e-26
+10.6953125, 5.77431219171125733219893180204e-26
+10.703125, 5.31127278202439078077610812502e-26
+10.7109375, 4.88506611742100396059158846966e-26
+10.71875, 4.49278646762238143463276383412e-26
+10.7265625, 4.13175539358415047424927076696e-26
+10.734375, 3.79950412322082429357326800941e-26
+10.7421875, 3.49375728147561788395406178234e-26
+10.75, 3.2124178716214397401778674819e-26
+10.7578125, 2.95355341245485167226228943449e-26
+10.765625, 2.7153831432400615202588824245e-26
+10.7734375, 2.49626621491758297034406488327e-26
+10.78125, 2.29469079225183755784603901466e-26
+10.7890625, 2.10926399729054980138026315718e-26
+10.796875, 1.93870262978046018993340371957e-26
+10.8046875, 1.78182460506031055457172102147e-26
+10.8125, 1.63754105446257713902047954594e-26
+10.8203125, 1.50484903742721884265768605182e-26
+10.828125, 1.38282481838895404534317752042e-26
+10.8359375, 1.27061766506760881921285620726e-26
+10.84375, 1.16744412809050399576203679864e-26
+10.8515625, 1.07258276492668755615447880825e-26
+10.859375, 9.8536927393361460194545761832e-27
+10.8671875, 9.05192006924803330713505393064e-27
+10.875, 8.31487831077956685494020064641e-27
+10.8828125, 7.63738313231764232625628711201e-27
+10.890625, 7.01466201679725828310821400957e-27
+10.8984375, 6.44232182473495976013551237824e-27
+10.90625, 5.91631889008127450680637373772e-27
+10.9140625, 5.43293145288019222635552789189e-27
+10.921875, 4.98873424775367552758882807386e-27
+10.9296875, 4.58057508111756165983390341723e-27
+10.9375, 4.20555324286787447664590645616e-27
+10.9453125, 3.86099961013298233605596874821e-27
+10.953125, 3.54445831164075439490824563329e-27
+10.9609375, 3.25366983136894493459071480682e-27
+10.96875, 2.98655543949445301107516124079e-27
+10.9765625, 2.74120284729113821635178734867e-27
+10.984375, 2.51585299060044114864697816635e-27
+10.9921875, 2.30888785386404327099028990864e-27
+11.0, 2.11881925350935339354284050142e-27
+11.0078125, 1.94427850575941637274956864704e-27
+11.015625, 1.78400690973839081264603967175e-27
+11.0234375, 1.63684698209855906651355275628e-27
+11.03125, 1.50173438433869289489738389171e-27
+11.0390625, 1.37769048854773979204248276621e-27
+11.046875, 1.26381553152110520247533138965e-27
+11.0546875, 1.15928231108599561918115084381e-27
+11.0625, 1.06333038206205418063048641813e-27
+11.0703125, 9.75260712596697721074623246501e-28
+11.078125, 8.94430764672261865509909072845e-28
+11.0859375, 8.20249965403792907758797324325e-28
+11.09375, 7.52175538350130054414232645958e-28
+11.1015625, 6.89708666463471842211742784666e-28
+11.109375, 6.32390960519325651296579238688e-28
+11.1171875, 5.79801208913840319180455092842e-28
+11.125, 5.31552386602179264480150186384e-28
+11.1328125, 4.87288902691966081266718796058e-28
+11.140625, 4.46684067811162717247660848708e-28
+11.1484375, 4.09437763850403747003071213627e-28
+11.15625, 3.75274300045400114648498126567e-28
+11.1640625, 3.43940440624402426277797922927e-28
+11.171875, 3.15203590407062095283937001436e-28
+11.1796875, 2.8885012581190837422803885855e-28
+11.1875, 2.64683859717053061304283503523e-28
+11.1953125, 2.42524629529091165178132033861e-28
+11.203125, 2.22206998654440375038514663744e-28
+11.2109375, 2.03579062341054126790131171401e-28
+11.21875, 1.86501349571630258966052443229e-28
+11.2265625, 1.70845813346808135591122271786e-28
+11.234375, 1.56494902302730223818204989801e-28
+11.2421875, 1.43340707165735475978410155883e-28
+11.25, 1.31284176061540277820326906487e-28
+11.2578125, 1.20234393170453152416886466378e-28
+11.265625, 1.10107915657104612169605886866e-28
+11.2734375, 1.00828164205754080298071783586e-28
+11.28125, 9.23248628631384868803031255643e-29
+11.2890625, 8.45335242325221794716567420911e-29
+11.296875, 7.73949763773745363925080465164e-29
+11.3046875, 7.0854928083043260651941253326e-29
+11.3125, 6.48635693918481552500087434619e-29
+11.3203125, 5.93752045729823619686737769267e-29
+11.328125, 5.43479149151269718210647157836e-29
+11.3359375, 4.97432489382838830240559087984e-29
+11.34375, 4.55259378134158403634286329612e-29
+11.3515625, 4.16636339553476307768722161539e-29
+11.359375, 3.81266709172235704726150255127e-29
+11.3671875, 3.48878428647358393345088492376e-29
+11.375, 3.19222020463522110274972424613e-29
+11.3828125, 2.9206872802817397220571482522e-29
+11.390625, 2.67208807761435847629039217408e-29
+11.3984375, 2.44449960859393641289642343862e-29
+11.40625, 2.23615893399864656606591195114e-29
+11.4140625, 2.04544994371374638130801746683e-29
+11.421875, 1.87089122044988394605076944968e-29
+11.4296875, 1.71112489880576068034662358508e-29
+11.4375, 1.56490643869361453280782051265e-29
+11.4453125, 1.43109523868073890455848345265e-29
+11.453125, 1.30864602081212351911922520333e-29
+11.4609375, 1.19660092400977044773317455887e-29
+11.46875, 1.09408224823151700980937996553e-29
+11.4765625, 1.00028579625149614235717709105e-29
+11.484375, 9.14474764228120624190303467099e-30
+11.4921875, 8.35974136183570798714496759642e-30
+11.5, 7.64165541158720278429188436563e-30
+11.5078125, 6.98482535154602353272946422878e-30
+11.515625, 6.3840627304925086850567681092e-30
+11.5234375, 5.83461538508546899815891459141e-30
+11.53125, 5.33213102511369228044918872847e-30
+11.5390625, 4.87262383501117231330713235114e-30
+11.546875, 4.45244384374339427572930921654e-30
+11.5546875, 4.06824883538218242549715556768e-30
+11.5625, 3.71697859126263659003568067665e-30
+11.5703125, 3.39583127168856982303130586879e-30
+11.578125, 3.10224176084321507981014348424e-30
+11.5859375, 2.83386181298080588733764856439e-30
+11.59375, 2.58854185122413007911934214361e-30
+11.6015625, 2.36431428246734260614209132562e-30
+11.609375, 2.15937820306874225303922446886e-30
+11.6171875, 1.97208538029450232676323566869e-30
+11.625, 1.80092740391477307929261321075e-30
+11.6328125, 1.64452391102556450778536093426e-30
+11.640625, 1.50161179513540141550651277058e-30
+11.6484375, 1.37103531787198317378812574666e-30
+11.65625, 1.25173704838347303201269249931e-30
+11.6640625, 1.14274956167987301782373568317e-30
+11.671875, 1.04318783282662408725473004371e-30
+11.6796875, 9.52242269105955523846497808342e-31
+11.6875, 8.69172327039159317422198730386e-31
+11.6953125, 7.93300665549430751257352943562e-31
+11.703125, 7.2400779057196247833329807759e-31
+11.7109375, 6.60727150114814871539924381072e-31
+11.71875, 6.02940642167565572289057832921e-31
+11.7265625, 5.50174500969555427457208925577e-31
+11.734375, 5.01995530008393330375319622108e-31
+11.7421875, 4.58007652743122665025754375567e-31
+11.75, 4.17848754454264531513339136318e-31
+11.7578125, 3.81187790832463913360545120591e-31
+11.765625, 3.47722140944873809374510806976e-31
+11.7734375, 3.17175184078606149142649941023e-31
+11.78125, 2.89294081667237175740254912079e-31
+11.7890625, 2.63847747072049136417835719221e-31
+11.796875, 2.40624987425969168355412206992e-31
+11.8046875, 2.19432803065641333502236747923e-31
+11.8125, 2.00094831285487117218095763464e-31
+11.8203125, 1.82449922255928343559024119791e-31
+11.828125, 1.66350835964390078579515562251e-31
+11.8359375, 1.51663049969825003375965839346e-31
+11.84375, 1.38263668616244035917623624956e-31
+11.8515625, 1.26040425134473379230227135993e-31
+11.859375, 1.14890768779937846515542856271e-31
+11.8671875, 1.04721029813068055754002960031e-31
+11.875, 9.54456557328790894520635007692e-32
+11.8828125, 8.69865127279007299944901406158e-32
+11.890625, 7.92722468161135091122668667415e-32
+11.8984375, 7.22376996106786573092446585927e-32
+11.90625, 6.582337407454844088562188271e-32
+11.9140625, 5.9974946017723384628909200415e-32
+11.921875, 5.46428174489372040391902186547e-32
+11.9296875, 4.97817082216088130824636574093e-32
+11.9375, 4.53502827144904283492152918513e-32
+11.9453125, 4.13108085628436501512415846805e-32
+11.953125, 3.76288447082854804981400650949e-32
+11.9609375, 3.42729562665848477478861626432e-32
+11.96875, 3.12144539244200812941311493537e-32
+11.9765625, 2.84271557700460538625113060076e-32
+11.984375, 2.5887169640449851832450877125e-32
+11.9921875, 2.35726942302566030679106783504e-32
+12.0, 2.14638373566306018809237666214e-32
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/xsqrt2pi.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/xsqrt2pi.csv
new file mode 100644
index 0000000..9f3807f
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/xsqrt2pi.csv
@@ -0,0 +1,527 @@
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements. See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+# High-precision test data for: x * sqrt(2 * pi)
+# Computed from uniform double in the range [1, 2) using:
+#
+# BigDecimal SQRT2PI = new BigDecimal("2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266116009972665203879644866323618");
+# MathContext mc = new MathContext(25);
+# for (int i = 0; i < 500; i++) {
+# // Combine an unbiased exponent of 0 with random 52 bit mantissa
+# double u = Double.longBitsToDouble(
+# (1023L << 52) | (ThreadLocalRandom.current().nextLong() >>> 12));
+# System.out.printf("%s,%s%n", u, new BigDecimal(u).multiply(SQRT2PI, mc));
+# }
+
+1.8893551562890456,4.735911055573994672337905
+1.5024003177840584,3.765959116372121064332974
+1.4456830404612515,3.623789985374685765626532
+1.1511923788242675,2.885611366300630771642443
+1.546677556686121,3.876945695326623102197423
+1.1693238567311188,2.931060241482791608388387
+1.058466759912108,2.653182708152752812829678
+1.038957669113387,2.604280669544335302822578
+1.8918931026719994,4.742272743737004126785173
+1.6489839885519744,4.133389890118180991453160
+1.7330476615361827,4.344106269689731841149198
+1.0838381218705355,2.716779281403644501968250
+1.1540296592857102,2.892723373728341173168258
+1.8956705935812812,4.751741509257371481861298
+1.87579736657157,4.701926716526668696994660
+1.0510503971641891,2.634592643593899226906572
+1.60008524086321,4.010818906567477001498907
+1.775523478397025,4.450577353221167439406124
+1.4384380398051464,3.605629461880372531786691
+1.5422781805915367,3.865918094817202122659544
+1.6541287014980426,4.146285773053655744024860
+1.8563919167106442,4.653284467223338024074239
+1.6462973604126097,4.126655512060630131567187
+1.3632198351705267,3.417085383376254225471836
+1.043488685448278,2.615638243202188155476037
+1.0175591388046203,2.550642508436832245150859
+1.771568154358829,4.440662826151697717963353
+1.0271355815300311,2.574647090542731328210567
+1.2349687890221102,3.095607684849628060205599
+1.0172867848070084,2.549959818205709371211783
+1.8292999923424442,4.585375083587843400036143
+1.474282039246478,3.695477044355871961188896
+1.6327763243141156,4.092763300673838421064559
+1.736345430789484,4.352372551343265300153152
+1.7065491254180385,4.277684289819660711199551
+1.4871741526658877,3.727792780372714234881868
+1.8688808855500951,4.684589669637291268499749
+1.6340563558185295,4.095971863835220795055308
+1.809461458322361,4.535647253285873623707234
+1.2153438419897076,3.046415237730071996779515
+1.0364918500304918,2.598099777711025445358362
+1.0561567113558583,2.647392275125886738609376
+1.1992814280074178,3.006152736683236050875163
+1.7653227746726068,4.425007980844406823447594
+1.2797628300133566,3.207889694533266329575205
+1.2775181449512887,3.202263103489045152089193
+1.8208481729073263,4.564189514019700888763745
+1.6417324221439529,4.115212908724469955783152
+1.4072565921878273,3.527469163638875000647085
+1.4634897951718762,3.668424900211756273760738
+1.7632530092447944,4.419819848301198677258669
+1.3238970655246862,3.318517817145188844829469
+1.541405500804272,3.863730610987745496224320
+1.4580100043270914,3.654689101541154642811058
+1.691317409187561,4.239504039245189668716656
+1.0798017660922488,2.706661637883320691941023
+1.8591920194865446,4.660303284013282644806625
+1.6313052018689096,4.089075743557240830447782
+1.8325804591704598,4.593597994492936191327527
+1.0884804288866146,2.728415819429666092236296
+1.9190100970266686,4.810244968509427135113695
+1.0608153098575288,2.659069649850327619905793
+1.7046365103027081,4.272890074693086991932291
+1.9544417188272403,4.899058873530772290660455
+1.6455787925619683,4.124854329568971587792795
+1.661846426502339,4.165631240765251633929839
+1.6135080120205985,4.044464804274488333582625
+1.6665022641414409,4.177301695033515785663458
+1.0992683901434253,2.755457228141611567865437
+1.469947897410839,3.684612961884398514575027
+1.3530044982905804,3.391479331118100138231124
+1.4545607787339854,3.646043175143894238195471
+1.271905085739578,3.188193250561793026510916
+1.5534999344176375,3.894046860248654999054064
+1.9655736053969848,4.926962375156478976081324
+1.3484497064622887,3.380062161136245756031231
+1.8771134816286847,4.705225727741500297590670
+1.329940216328096,3.333665749816874846882118
+1.214238061413032,3.043643456871039033630039
+1.7238474614133552,4.321044787929588781692988
+1.412761669903627,3.541268347095339488393443
+1.547261761209633,3.878410078903425723078162
+1.938916659577777,4.860143321050745926587406
+1.275452420018462,3.197085098964811671377711
+1.3593106809356277,3.407286586841162838843354
+1.9859751903494856,4.978101564845703771257382
+1.5207738329035645,3.812014688875035326359163
+1.9504551632210487,4.889066060529903838060832
+1.2640747030704744,3.168565391962237583069059
+1.8347902658610418,4.599137158425017833390418
+1.774248682608293,4.447381914052751232248882
+1.9139983676499444,4.797682425948931384726939
+1.9794545170713098,4.961756660836997653504486
+1.9679867700096736,4.933011281805983718915616
+1.243301495722973,3.116494683070218323694455
+1.7741879105038194,4.447229580977368780424269
+1.3844462785358902,3.470292186485727985691636
+1.5877648602381693,3.979936292138534010265939
+1.2293461719711378,3.081513873972238434712924
+1.9249068577691122,4.825025955715170372473915
+1.9903075572339244,4.988961198174313163574148
+1.8817474014295563,4.716841242136737449621304
+1.0842697581794554,2.717861233179940352004540
+1.1938308028474758,2.992490045542910337281433
+1.506808273053191,3.777008221683037879468801
+1.9312368918062757,4.840892998012100970101116
+1.922592342128337,4.819224325367927777339472
+1.5697414972237322,3.934758420802607411465733
+1.9701326714821867,4.938390259111557467758925
+1.035534629954743,2.595700382804108993743693
+1.3129433742279952,3.291060984829323695101888
+1.2626367359606856,3.164960942946851483823105
+1.945918113196598,4.877693362655200147559894
+1.7419987849376937,4.366543408697670383243248
+1.4525799996169642,3.641078098203370235607379
+1.2597206791116362,3.157651472398592908120723
+1.0951520868774027,2.745139185988043573337873
+1.6740434131335336,4.196204552320300376856637
+1.2674056570446908,3.176914855375502771797759
+1.7088681586327104,4.283497244045365622598640
+1.6054641455675047,4.024301821185807786801248
+1.5329945253797894,3.842647422171511061311788
+1.2084837793832828,3.029219610835068757673590
+1.6430911993943307,4.118618858199192333814614
+1.8181849928285725,4.557513911533862677994276
+1.2016570102994337,3.012107438425115959668259
+1.366668624756172,3.425730216864885432971487
+1.7066520521791089,4.277942288949175906454654
+1.1872551468274466,2.976007320238857627570212
+1.559576801534092,3.909279307183935480142678
+1.672450102921241,4.192210715891909735052209
+1.46762611384187,3.678793113542846992545807
+1.6387351332392013,4.107699819608581826965633
+1.5614014900511697,3.913853123013236757911259
+1.0232739134726612,2.564967324202888361711284
+1.2372585212835128,3.101347192477394624907342
+1.0223451836497681,2.562639343769331604861555
+1.995062378164493,5.000879766759683600332344
+1.2977571613231147,3.252994794177383962746311
+1.4899046307352726,3.734637073904694210247428
+1.645552132057302,4.124787501594159474745513
+1.8651228301268465,4.675169621655745993011740
+1.917082187504684,4.805412415990690131059641
+1.2474506511032901,3.126895073262358297171502
+1.8662329980373273,4.677952399929745176476855
+1.7983000488242025,4.507669748653054597012085
+1.0762470665334183,2.697751327461337876238052
+1.5243571274151704,3.820996676214156715507868
+1.8395294863887859,4.611016622599572883066556
+1.8279119236839136,4.581895711441241516178514
+1.0483050802541538,2.627711154604382076785142
+1.326220733229557,3.324342388315064920373633
+1.5132555903700033,3.793169249564877544002960
+1.1442261047763294,2.868149506803240856105877
+1.4405360401644791,3.610888368901261924287071
+1.0767297924828614,2.698961341975110037748288
+1.8620023879008947,4.667347832942822540060959
+1.0620558592669116,2.662179246075963297298955
+1.0413309641262987,2.610229637927740299157990
+1.3958324841630558,3.498833171451543971173777
+1.3886651588131336,3.480867351075949304652253
+1.6916233770104616,4.240270986841199879243101
+1.3362451351502092,3.349469837605636847092560
+1.51007213672281,3.785189514641845632353889
+1.979964958419879,4.963036147553862131692244
+1.6647526145913876,4.172915974000656774745694
+1.3137029364162183,3.292964924886664160277590
+1.3675438301973761,3.427924031569918855140226
+1.4306993069238463,3.586231335230289034851947
+1.3086292668249073,3.280247121232948626426439
+1.4703463824452223,3.685611815738600806425211
+1.4495104084027266,3.633383774074203475879939
+1.8410364103686825,4.614793920855301182149402
+1.4175646545073626,3.553307644105680583225269
+1.9184532391665718,4.808849132852358138262893
+1.9927468765970628,4.995075665060810910264185
+1.1312281494145473,2.835568464381006343558583
+1.9218725610063763,4.817420101656075198363312
+1.0994865856226224,2.756004163099163686586501
+1.6114310008331842,4.039258509305390959835820
+1.8221200517604763,4.567377641514912065951538
+1.7548403471909473,4.398732431732110071154756
+1.0053514819852616,2.520042450686435632502894
+1.536400087661288,3.851183900877332248801775
+1.9984199735214154,5.009296010216115261114838
+1.898948155903,4.759957139644856850309903
+1.1698728161374878,2.932436278652420817010494
+1.1770798591343048,2.950501656404723546560846
+1.4540065074682047,3.644653823117272895843646
+1.237258978804093,3.101348339311417456399954
+1.2432279677488296,3.116310375771253875037401
+1.8118933038119949,4.541742985949723908658576
+1.396339943427966,3.500105183193191184360900
+1.8243693990351244,4.573015918993009162778822
+1.3883078091733525,3.479971608364944875973750
+1.8147314952913527,4.548857276960699010698276
+1.6884685291486183,4.232362955988544358284171
+1.210105722038836,3.033285218155308663136598
+1.1436158645086882,2.866619861294053246522626
+1.1109011974447085,2.784616351836342089410470
+1.7409917163546644,4.364019062112956723303313
+1.029834808379005,2.581413048882012283632299
+1.6751874066538013,4.199072118824198115404159
+1.8250103532092061,4.574622552848505163515521
+1.6305453511281864,4.087171080206044858619874
+1.3890598429812357,3.481856677571263437162032
+1.159447696616209,2.906304379293975825241757
+1.7677911511316267,4.431195283069019583755771
+1.8850697031318697,4.725169017520610927338587
+1.1664568129345874,2.923873628437800427858445
+1.5699383564352898,3.935251873668419114425110
+1.8712499502159996,4.690528034113276669082218
+1.0478756440573032,2.626634717691206227889433
+1.7705431957109004,4.438093635824472000065583
+1.6323018994163372,4.091574093810878210410434
+1.2768483900129608,3.200584276823558740473157
+1.8891265160854618,4.735337939574974084306633
+1.8354687817874407,4.600837945630916744695152
+1.3091986735212537,3.281674412157774789940609
+1.0810129136258926,2.709697534535901890420026
+1.4249694986337922,3.571868835762224336040193
+1.6184009923546407,4.056729687127012163052258
+1.6737532395102033,4.195477194911508568857607
+1.9111506134937,4.790544164861691257728156
+1.5002649681761948,3.760606588668828080809955
+1.6521903396002537,4.141427020314190825187225
+1.2241879911729283,3.068584232137787771946601
+1.9309471768937783,4.840166790420852722405790
+1.921886044262323,4.817453899166665116662880
+1.014046777985023,2.541838325495723610704574
+1.8754346638370598,4.701017555597059669299899
+1.6857858714713396,4.225638530403521617337409
+1.0616412270919842,2.661139917342718449834139
+1.932175852487611,4.843246623404702708208100
+1.081302438638131,2.710423266117791385270858
+1.7708439273549381,4.438847458266493329884588
+1.8244741558360644,4.573278505352205308674198
+1.2690481736248025,3.181032033876761021993632
+1.451802661925381,3.639129601566711472704383
+1.0407715272442646,2.608827337621362263759637
+1.6161358727446726,4.051051874267244945455029
+1.815938243737279,4.551882146735625110440865
+1.0701088828743806,2.682365182746715923213719
+1.1421936355577804,2.863054861992708913972317
+1.515418424313111,3.798590670280002906861926
+1.7687352853595328,4.433561876619736121757592
+1.0071332637373647,2.524508715205478787791024
+1.9482404066001433,4.883514488962516093802804
+1.6566683983128427,4.152651848898624096576961
+1.4651879105005756,3.672681444108258646906179
+1.2443682629343111,3.119168671924607485326056
+1.2626431956837838,3.164977135071415735548666
+1.4147099189897463,3.546151883340630250384757
+1.4414804634096674,3.613255686910869648299407
+1.1438586674197226,2.867228477936014791395197
+1.07076682719242,2.684014404577446418841217
+1.3562607705914937,3.399641595337467158429472
+1.1289837700421548,2.829942639587168662674399
+1.427408079858172,3.577981452409239274394226
+1.9857843698367643,4.977623248753137139608671
+1.6706063378605387,4.187589082258976394402244
+1.0406539599737064,2.608532640176809817745586
+1.2706995074119407,3.185171313838455175603385
+1.1367365708176882,2.849376029218701880352147
+1.0732776182050614,2.690308024321422754255388
+1.3802065574461688,3.459664781725683040042045
+1.9626896669490403,4.919733413500566013769454
+1.4014644898649418,3.512950516186774266252136
+1.5883682772508232,3.981448834283845580367604
+1.7526988175409448,4.393364412960453245284838
+1.4091466227593312,3.532206767709323771282161
+1.4335812948920779,3.593455407758604664834706
+1.0977083215951666,2.751546716208183946582026
+1.6448225193196415,4.122958633676408509553359
+1.5370323883498,3.852768843661225215486209
+1.090863069420763,2.734388213560844618783650
+1.7065314667461202,4.277640026093337931266329
+1.7068930070889805,4.278546273339171318453332
+1.0041079740576166,2.516925438555273030818284
+1.9402614453060345,4.863514198980516618741448
+1.4654440615642594,3.673323519607065311743763
+1.393593500487687,3.493220871664427050132880
+1.4160474273166892,3.549504519530499739079956
+1.0895661560101162,2.731137333735969072615490
+1.4676998622716726,3.678977973442199746564850
+1.3052249447555522,3.271713751277952694773496
+1.0519329367473378,2.636804842266500704883524
+1.1595057307386436,2.906449849266163658243075
+1.0388877325485937,2.604105364573593675088588
+1.0818218936700918,2.711725346788304017242256
+1.3608460722162794,3.411135242037866344885318
+1.2602777135679555,3.159047750716746502191915
+1.10899836625668,2.779846661378580215231556
+1.102336984879933,2.763149054471525553889700
+1.3723747048151416,3.440033238478006921320639
+1.5968993312649995,4.002833015488184213079601
+1.5877866872414583,3.979991004322128706560639
+1.3307162560442987,3.335610992911745048996039
+1.3616570684941565,3.413168108238613490669377
+1.467452662934309,3.678358336593694067270730
+1.8182304135826795,4.557627764480362250531911
+1.9003350705393278,4.763433619086775926124495
+1.4047971225333544,3.521304187462376400881233
+1.5591335294601796,3.908168188870112130557393
+1.3373727308733545,3.352296300927625991546563
+1.5585280450508483,3.906650464529834092307228
+1.890684258788732,4.739242621479591417740427
+1.1083312235020393,2.778174382486582503575470
+1.9962311488798643,5.003809440481394149878685
+1.349594795591025,3.382932473923308850958347
+1.3973482510387782,3.502632635559978865375806
+1.392846796796923,3.491349163080386642631921
+1.365302361843278,3.422305503616845925181124
+1.3472542896539814,3.377065695564573633534791
+1.7826814840816936,4.468519812660326963563345
+1.1262306362515802,2.823041556583872285430067
+1.352893129889564,3.391200171935211600428665
+1.3669397184442964,3.426409747968612307600785
+1.7984909862147127,4.508148357714791818935931
+1.9277770576850048,4.832220479978190426590575
+1.808924806628063,4.534302066975317746246027
+1.3129460913823288,3.291067795725202660982419
+1.582738065873675,3.967335987253736625326882
+1.4801987567456454,3.710308055732289102958679
+1.552777753699793,3.892236621641913023016809
+1.1642820533244869,2.918422314508597033614973
+1.6158189882098521,4.050257562532470631065407
+1.29643704859647,3.249685762291076199774175
+1.6239268687978277,4.070581005261621874009168
+1.3316665570168904,3.337993044199052854952387
+1.1714888962177525,2.936487190675680259578640
+1.3672696358590695,3.427236728288775628494555
+1.6517809379083743,4.140400802457644105777844
+1.8018171647215644,4.516485850806536174843265
+1.8778101425923872,4.706971997810948581946490
+1.058526239733344,2.653331801954432682150976
+1.363790111323167,3.418514853704810110504352
+1.1477802493768323,2.877058426150988682285753
+1.6732476364778937,4.194209836054982102674285
+1.778073852157181,4.456970192199251421650165
+1.976802741233993,4.955109644545196080637326
+1.7525229677976533,4.392923623021832055175056
+1.0585840834646543,2.653476794686845357634822
+1.9456818646853415,4.877101175457045462772195
+1.7784531953066671,4.457921064463540770531805
+1.0780010623734697,2.702147943027595933274805
+1.45722649975433,3.652725146825768414038107
+1.0646641652352042,2.668717299564974432932586
+1.189189545626613,2.980856138963260528123878
+1.7141399218412916,4.296711594761154824121746
+1.831191839333279,4.590117240746345151341961
+1.5105096570855083,3.786286215553711963709925
+1.9294142858601775,4.836324402414100694527398
+1.556876974689014,3.902511844877455234898603
+1.5896693241142186,3.984710075138252424544220
+1.4895771369299708,3.733816168672558403431955
+1.792805213210113,4.493896238338328794998684
+1.9923670756125573,4.994123645174316687428117
+1.3730767256298864,3.441792943701625721188289
+1.4014904000431603,3.513015463372097520176136
+1.5222236628728323,3.815648873669409365120058
+1.590400391995625,3.986542590560460623261960
+1.6008419285300874,4.012715641268336506773172
+1.6544505680237191,4.147092572787573858803903
+1.11813847545662,2.802757517532364514549537
+1.9255372169083143,4.826606031756666323689313
+1.4477028001965109,3.628852772235048109688925
+1.809563770409103,4.535903711655337748963729
+1.8888006261667238,4.734521054690248358016730
+1.4622708766160477,3.665369524495244290013654
+1.2956226183650252,3.247644288445222432238724
+1.502325972142849,3.765772759485770163345473
+1.6247987407722884,4.072766464204663429350821
+1.2732060830874377,3.191454367299158122570425
+1.3736718859256334,3.443284789326882953553161
+1.4877801595192106,3.729311814285873589042828
+1.8045744032952535,4.523397222975248641955513
+1.2537403670938796,3.142661053203768601664663
+1.5702026332502963,3.935914317405243869852487
+1.2202667825013327,3.058755219610837920984719
+1.5390959168018543,3.857941342424649974628027
+1.2327898396456551,3.090145868733616320526092
+1.642847475638893,4.118007933342613257637439
+1.843148606113964,4.620088410431978915904332
+1.8638008121546559,4.671855814027082519952633
+1.4605498189046031,3.661055472573465544668716
+1.4077344238363243,3.528666909959511167530833
+1.1976968105794263,3.002180689833759677859805
+1.5578433494730552,3.904934187235023226985103
+1.1777034924088066,2.952064873203590391836703
+1.1301087507133416,2.832762547945978921197835
+1.852561182602557,4.643682240795413260905273
+1.4835305542826804,3.718659633643967094280204
+1.008925822791071,2.529001994413444621657194
+1.2435005300494313,3.116993588140540536334612
+1.7982627147260009,4.507576165946894582261155
+1.337227524892358,3.351932323510014524216890
+1.8632570946609264,4.670492916383888649415488
+1.2722820393319392,3.189138133094629411020291
+1.3770635563710492,3.451786446363592449887228
+1.3898403938651984,3.483813228486792491467174
+1.8202015058272696,4.562568560032557763017565
+1.251665543157049,3.137460240858827634305016
+1.7169876311265178,4.303849743373431933389685
+1.1799741448365777,2.957756554780901046861935
+1.0128333293104017,2.538796660738104185151900
+1.4260083233476673,3.574472783162409030284279
+1.9006829212890473,4.764305551611374256170075
+1.3326770890761723,3.340526072431270008845053
+1.2610940584776253,3.161094023949176072943452
+1.2951856093852274,3.246548869380193581608742
+1.9342252431404892,4.848383683960971779927945
+1.155073771406277,2.895340574691639036092312
+1.155661204203503,2.896813050350611183435679
+1.3876538059179322,3.478332265313207606476319
+1.417943825081248,3.554258083787089709263072
+1.4765523800436582,3.701167944791132256634783
+1.281566489435526,3.212410798238681097339401
+1.0687751123409281,2.679021915815694451478285
+1.699089164835062,4.258984941694738805834986
+1.567269949713454,3.928563169931249876187599
+1.0893445450217565,2.730581837366577700489578
+1.237851395022049,3.102833306553695994130305
+1.1893698456378419,2.981308084069322969653260
+1.6294530841535018,4.084433172923854589753967
+1.8345607776359956,4.598561916751422136781891
+1.8039904714844286,4.521933522987778354991799
+1.3416713345140339,3.363071302354784620047780
+1.179685435792206,2.957032868527137180798707
+1.1889707074835496,2.980307593086289943023398
+1.6534662420776016,4.144625233539582758005152
+1.269601328926249,3.182418588595629147259615
+1.2390048988917177,3.105724711968303617189501
+1.411242049913086,3.537459224660355037945296
+1.8391754528149993,4.610129192033350717966608
+1.0128086464141064,2.538734789892350566710238
+1.599338684235736,4.008947566616437443833993
+1.546091397661165,3.875476412541238084519486
+1.7899483461869587,4.486735134681229197877519
+1.048513801145876,2.628234340293078839647520
+1.8605073020837144,4.663600208560478667887220
+1.9229671198347507,4.820163753763525430816229
+1.0075748864361922,2.525615699149078715883446
+1.8667446757585564,4.679234985773276708809141
+1.3984528612532663,3.505401482756060811269347
+1.7452909584235463,4.374795663842299125930015
+1.172610877703606,2.939299581191732911913693
+1.9575441226972359,4.906835446790627917599890
+1.142351159612969,2.863449716243379043501909
+1.108589993890696,2.778823023659426734122931
+1.3489182484554298,3.381236621744105375554393
+1.4546040086311394,3.646151536426409834091616
+1.3511776486152478,3.386900098068410873323576
+1.2510689893459848,3.135964902208675331205520
+1.4666707645655295,3.676398408034623490313268
+1.5204062587523923,3.811093317114683681542370
+1.4583592519234445,3.655564535441020314465431
+1.0147712749783413,2.543654370144060218954965
+1.9934266577339108,4.996779623678994831867913
+1.8226535460990105,4.568714913508237514490894
+1.1536029268187253,2.891653714060893831945991
+1.247178734589418,3.126213479640347551064355
+1.212511882093139,3.039316566980712358290312
+1.992577532383992,4.994651182068182508985168
+1.8035330090535662,4.520786834723997181566305
+1.1586592342454123,2.904327997221854044612603
+1.3954536405661617,3.497883551379906292749470
+1.6016271919801595,4.014684004835321310296238
+1.1173907408656671,2.800883224864762614145960
+1.3479041444986948,3.378694640092738080071069
+1.917209059360735,4.805730436572322433534557
+1.8270868901379358,4.579827659027374364168069
+1.9633021507081323,4.921268682608858185188936
+1.7459750201802204,4.376510352383172241255689
+1.4326446152417949,3.591107500062933844820993
+1.9142600727041912,4.798338423237520329140572
+1.8848065337188515,4.724509349628921506745030
+1.5539614179906223,3.895203628020976465293418
+1.989621971788895,4.987242690313127015115888
+1.4217725323173909,3.563855229600489876705864
+1.2939825189504992,3.243533168879565734109171
+1.0379608252237977,2.601781952465297579756487
+1.1969778392747337,3.000378496032768684398245
+1.592736428410693,3.992398165489037626061497
+1.7042088152907808,4.271818002283271383273116
+1.0865862996890705,2.723667941627298026511420
+1.9636717353971938,4.922195094040330329726462
+1.9525731650465763,4.894375103791491323247340
+1.5894574799689931,3.984179060614015252252987
+1.9537839333460534,4.897410049844987474256616
+1.8978051164084127,4.757091964528704464889649
+1.143174776898094,2.865514218617748274780349
+1.1711630449491277,2.935670402672420872487957
+1.135895666139741,2.847268193776690018323994
+1.8016633701554452,4.516100344998617201385030
+1.7488781471069224,4.383787412422485862823032
+1.7966066493281243,4.503425025595939185917507
+1.7232857563932942,4.319636802244301625913431
+1.7095967104211915,4.285323452557905559905488
+1.6390331192290146,4.108446759716091857848058