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From celes...@apache.org
Subject svn commit: r1379261 - in /commons/proper/math/trunk/src: main/java/org/apache/commons/math3/special/Gamma.java test/java/org/apache/commons/math3/special/GammaTest.java
Date Fri, 31 Aug 2012 02:48:30 GMT
Author: celestin
Date: Fri Aug 31 02:48:29 2012
New Revision: 1379261

URL: http://svn.apache.org/viewvc?rev=1379261&view=rev
Log:
MATH-849: implementation of double Gamma.gamma(double).

Modified:
    commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
    commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java

Modified: commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java?rev=1379261&r1=1379260&r2=1379261&view=diff
==============================================================================
--- commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java (original)
+++ commons/proper/math/trunk/src/main/java/org/apache/commons/math3/special/Gamma.java Fri
Aug 31 02:48:29 2012
@@ -86,6 +86,10 @@ public class Gamma {
     };
     /** Avoid repeated computation of log of 2 PI in logGamma */
     private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
+
+    /** The constant value of √(2π). */
+    private static final double SQRT_TWO_PI = 2.506628274631000502;
+
     // limits for switching algorithm in digamma
     /** C limit. */
     private static final double C_LIMIT = 49;
@@ -623,4 +627,75 @@ public class Gamma {
 
         return -FastMath.log1p(invGamma1pm1(x));
     }
+
+
+    /**
+     * Returns the value of Γ(x). The present implementation is based on
+     * the double precision implementation in the
+     * <em>NSWC Library of Mathematics Subroutines</em>, {@code DGAMMA}.
+     *
+     * @param x the argument
+     * @return the value of {@code Gamma(x)}
+     */
+    public static double gamma(final double x) {
+
+        // TODO Check whether x is a negative integer
+
+        final double ret;
+        final double absX = FastMath.abs(x);
+        if (absX <= 20.0) {
+            if (x >= 1.0) {
+                /*
+                 * From the recurrence relation
+                 * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
+                 * then
+                 * Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
+                 * where t = x - n. This means that t must satisfy
+                 * -0.5 <= t - 1 <= 1.5.
+                 */
+                double prod = 1.0;
+                double t = x;
+                while (t > 2.5) {
+                    t = t - 1.0;
+                    prod *= t;
+                }
+                ret = prod / (1.0 + invGamma1pm1(t - 1.0));
+            } else {
+                /*
+                 * From the recurrence relation
+                 * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
+                 * then
+                 * Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
+                 * which requires -0.5 <= x + n <= 1.5.
+                 */
+                double prod = x;
+                double t = x;
+                while (t < -0.5) {
+                    t = t + 1.0;
+                    prod *= t;
+                }
+                ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
+            }
+        } else {
+            final double y = absX + LANCZOS_G + 0.5;
+            final double gammaAbs = SQRT_TWO_PI / x *
+                                    FastMath.pow(y, absX + 0.5) *
+                                    FastMath.exp(-y) * lanczos(absX);
+            if (x > 0.0) {
+                ret = gammaAbs;
+            } else {
+                /*
+                 * From the reflection formula
+                 * Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
+                 * and the recurrence relation
+                 * Gamma(1 - x) = -x * Gamma(-x),
+                 * it is found
+                 * Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
+                 */
+                ret = -FastMath.PI /
+                      (x * FastMath.sin(FastMath.PI * x) * gammaAbs);
+            }
+        }
+        return ret;
+    }
 }

Modified: commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java?rev=1379261&r1=1379260&r2=1379261&view=diff
==============================================================================
--- commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
(original)
+++ commons/proper/math/trunk/src/test/java/org/apache/commons/math3/special/GammaTest.java
Fri Aug 31 02:48:29 2012
@@ -445,6 +445,515 @@ public class GammaTest {
         Gamma.logGamma1p(1.51);
     }
 
+    /**
+     * Reference data for the {@link GammaNSWC#gamma(double)} function. This
+     * data was generated with the following <a
+     * href="http://maxima.sourceforge.net/">Maxima</a> script.
+     *
+     * <pre>
+     * kill(all);
+     *
+     * fpprec : 64;
+     *
+     * EPSILON : 10**(-fpprec + 1);
+     * isInteger(x) := abs(x - floor(x)) <= EPSILON * abs(x);
+     *
+     * x : makelist(bfloat(i / 8), i, -160, 160);
+     * x : append(x, makelist(bfloat(i / 2), i, 41, 200));
+     *
+     * for i : 1 while i <= length(x) do if not(isInteger(x[i])) then
+     *     print("{", float(x[i]), ",", float(gamma(x[i])), "},");
+     * </pre>
+     */
+    private final static double[][] GAMMA_REF = {
+        { - 19.875 , 4.920331854832504e-18 },
+        { - 19.75 , 3.879938752480031e-18 },
+        { - 19.625 , 4.323498423815027e-18 },
+        { - 19.5 , 5.811045977502237e-18 },
+        { - 19.375 , 9.14330910942125e-18 },
+        { - 19.25 , 1.735229114436739e-17 },
+        { - 19.125 , 4.653521565668223e-17 },
+        { - 18.875 , - 9.779159561479603e-17 },
+        { - 18.75 , - 7.662879036148061e-17 },
+        { - 18.625 , - 8.484865656736991e-17 },
+        { - 18.5 , - 1.133153965612936e-16 },
+        { - 18.375 , - 1.771516139950367e-16 },
+        { - 18.25 , - 3.340316045290721e-16 },
+        { - 18.125 , - 8.899859994340475e-16 },
+        { - 17.875 , 1.845816367229275e-15 },
+        { - 17.75 , 1.436789819277761e-15 },
+        { - 17.625 , 1.580306228567265e-15 },
+        { - 17.5 , 2.096334836383932e-15 },
+        { - 17.375 , 3.255160907158799e-15 },
+        { - 17.25 , 6.096076782655566e-15 },
+        { - 17.125 , 1.613099623974211e-14 },
+        { - 16.875 , - 3.29939675642233e-14 },
+        { - 16.75 , - 2.550301929218027e-14 },
+        { - 16.625 , - 2.785289727849803e-14 },
+        { - 16.5 , - 3.66858596367188e-14 },
+        { - 16.375 , - 5.655842076188414e-14 },
+        { - 16.25 , - 1.051573245008085e-13 },
+        { - 16.125 , - 2.762433106055837e-13 },
+        { - 15.875 , 5.567732026462681e-13 },
+        { - 15.75 , 4.271755731440195e-13 },
+        { - 15.625 , 4.630544172550298e-13 },
+        { - 15.5 , 6.053166840058604e-13 },
+        { - 15.375 , 9.261441399758529e-13 },
+        { - 15.25 , 1.708806523138138e-12 },
+        { - 15.125 , 4.454423383515037e-12 },
+        { - 14.875 , - 8.838774592009505e-12 },
+        { - 14.75 , - 6.728015277018307e-12 },
+        { - 14.625 , - 7.235225269609841e-12 },
+        { - 14.5 , - 9.382408602090835e-12 },
+        { - 14.375 , - 1.423946615212874e-11 },
+        { - 14.25 , - 2.605929947785661e-11 },
+        { - 14.125 , - 6.737315367566492e-11 },
+        { - 13.875 , 1.314767720561414e-10 },
+        { - 13.75 , 9.923822533602004e-11 },
+        { - 13.625 , 1.058151695680439e-10 },
+        { - 13.5 , 1.360449247303171e-10 },
+        { - 13.375 , 2.046923259368506e-10 },
+        { - 13.25 , 3.713450175594567e-10 },
+        { - 13.125 , 9.516457956687671e-10 },
+        { - 12.875 , - 1.8242402122789617e-9 },
+        { - 12.75 , - 1.3645255983702756e-9 },
+        { - 12.625 , - 1.4417316853645984e-9 },
+        { - 12.5 , - 1.836606483859281e-9 },
+        { - 12.375 , - 2.7377598594053765e-9 },
+        { - 12.25 , - 4.9203214826628017e-9 },
+        { - 12.125 , - 1.2490351068152569e-8 },
+        { - 11.875 , 2.3487092733091633e-8 },
+        { - 11.75 , 1.7397701379221012e-8 },
+        { - 11.625 , 1.8201862527728055e-8 },
+        { - 11.5 , 2.295758104824101e-8 },
+        { - 11.375 , 3.3879778260141535e-8 },
+        { - 11.25 , 6.027393816261931e-8 },
+        { - 11.125 , 1.5144550670134987e-7 },
+        { - 10.875 , - 2.7890922620546316e-7 },
+        { - 10.75 , - 2.044229912058469e-7 },
+        { - 10.625 , - 2.1159665188483867e-7 },
+        { - 10.5 , - 2.640121820547716e-7 },
+        { - 10.375 , - 3.8538247770911e-7 },
+        { - 10.25 , - 6.780818043294673e-7 },
+        { - 10.125 , - 1.6848312620525174e-6 },
+        { - 9.875 , 3.0331378349844124e-6 },
+        { - 9.75 , 2.1975471554628537e-6 },
+        { - 9.625 , 2.2482144262764103e-6 },
+        { - 9.5 , 2.772127911575102e-6 },
+        { - 9.375 , 3.998343206232017e-6 },
+        { - 9.25 , 6.95033849437704e-6 },
+        { - 9.125 , 1.7058916528281737e-5 },
+        { - 8.875 , - 2.9952236120471065e-5 },
+        { - 8.75 , - 2.1426084765762826e-5 },
+        { - 8.625 , - 2.163906385291045e-5 },
+        { - 8.5 , - 2.633521515996347e-5 },
+        { - 8.375 , - 3.748446755842515e-5 },
+        { - 8.25 , - 6.429063107298763e-5 },
+        { - 8.125 , - 1.5566261332057085e-4 },
+        { - 7.875 , 2.658260955691807e-4 },
+        { - 7.75 , 1.874782417004247e-4 },
+        { - 7.625 , 1.8663692573135265e-4 },
+        { - 7.5 , 2.238493288596895e-4 },
+        { - 7.375 , 3.1393241580181064e-4 },
+        { - 7.25 , 5.303977063521479e-4 },
+        { - 7.125 , .001264758733229638 },
+        { - 6.875 , - .002093380502607298 },
+        { - 6.75 , - .001452956373178292 },
+        { - 6.625 , - .001423106558701564 },
+        { - 6.5 , - .001678869966447671 },
+        { - 6.375 , - .002315251566538353 },
+        { - 6.25 , - .003845383371053072 },
+        { - 6.125 , - .009011405974261174 },
+        { - 5.875 , .01439199095542518 },
+        { - 5.75 , .009807455518953468 },
+        { - 5.625 , .009428080951397862 },
+        { - 5.5 , .01091265478190986 },
+        { - 5.375 , 0.014759728736682 },
+        { - 5.25 , 0.0240336460690817 },
+        { - 5.125 , .05519486159234969 },
+        { - 4.875 , - 0.0845529468631229 },
+        { - 4.75 , - .05639286923398244 },
+        { - 4.625 , - .05303295535161297 },
+        { - 4.5 , - .06001960130050425 },
+        { - 4.375 , - .07933354195966577 },
+        { - 4.25 , - .1261766418626789 },
+        { - 4.125 , - .2828736656607921 },
+        { - 3.875 , .4121956159577241 },
+        { - 3.75 , .2678661288614166 },
+        { - 3.625 , 0.24527741850121 },
+        { - 3.5 , .2700882058522691 },
+        { - 3.375 , .3470842460735378 },
+        { - 3.25 , .5362507279163854 },
+        { - 3.125 , 1.166853870850768 },
+        { - 2.875 , - 1.597258011836181 },
+        { - 2.75 , - 1.004497983230312 },
+        { - 2.625 , - .8891306420668862 },
+        { - 2.5 , - .9453087204829419 },
+        { - 2.375 , - 1.17140933049819 },
+        { - 2.25 , - 1.742814865728253 },
+        { - 2.125 , - 3.646418346408649 },
+        { - 1.875 , 4.59211678402902 },
+        { - 1.75 , 2.762369453883359 },
+        { - 1.625 , 2.333967935425576 },
+        { - 1.5 , 2.363271801207355 },
+        { - 1.375 , 2.782097159933201 },
+        { - 1.25 , 3.921333447888569 },
+        { - 1.125 , 7.748638986118379 },
+        { - 0.875 , - 8.610218970054413 },
+        { - 0.75 , - 4.834146544295877 },
+        { - 0.625 , - 3.792697895066561 },
+        { - 0.5 , - 3.544907701811032 },
+        { - 0.375 , - 3.825383594908152 },
+        { - 0.25 , - 4.901666809860711 },
+        { - 0.125 , - 8.717218859383175 },
+        { 0.125 , 7.533941598797612 },
+        { 0.25 , 3.625609908221908 },
+        { 0.375 , 2.370436184416601 },
+        { 0.5 , 1.772453850905516 },
+        { 0.625 , 1.434518848090557 },
+        { 0.75 , 1.225416702465178 },
+        { 0.875 , 1.089652357422897 },
+        { 1.0 , 1.0 },
+        { 1.125 , .9417426998497015 },
+        { 1.25 , 0.906402477055477 },
+        { 1.375 , .8889135691562253 },
+        { 1.5 , 0.886226925452758 },
+        { 1.625 , 0.896574280056598 },
+        { 1.75 , .9190625268488832 },
+        { 1.875 , .9534458127450348 },
+        { 2.0 , 1.0 },
+        { 2.125 , 1.059460537330914 },
+        { 2.25 , 1.133003096319346 },
+        { 2.375 , 1.22225615758981 },
+        { 2.5 , 1.329340388179137 },
+        { 2.625 , 1.456933205091972 },
+        { 2.75 , 1.608359421985546 },
+        { 2.875 , 1.78771089889694 },
+        { 3.0 , 2.0 },
+        { 3.125 , 2.251353641828193 },
+        { 3.25 , 2.549256966718529 },
+        { 3.375 , 2.902858374275799 },
+        { 3.5 , 3.323350970447843 },
+        { 3.625 , 3.824449663366426 },
+        { 3.75 , 4.422988410460251 },
+        { 3.875 , 5.139668834328703 },
+        { 4.0 , 6.0 },
+        { 4.125 , 7.035480130713102 },
+        { 4.25 , 8.28508514183522 },
+        { 4.375 , 9.797147013180819 },
+        { 4.5 , 11.63172839656745 },
+        { 4.625 , 13.86363002970329 },
+        { 4.75 , 16.58620653922594 },
+        { 4.875 , 19.91621673302373 },
+        { 5.0 , 24.0 },
+        { 5.125 , 29.02135553919155 },
+        { 5.25 , 35.21161185279968 },
+        { 5.375 , 42.86251818266609 },
+        { 5.5 , 52.34277778455352 },
+        { 5.625 , 64.11928888737773 },
+        { 5.75 , 78.78448106132322 },
+        { 5.875 , 97.09155657349066 },
+        { 6.0 , 120.0 },
+        { 6.125 , 148.7344471383567 },
+        { 6.25 , 184.8609622271983 },
+        { 6.375 , 230.3860352318302 },
+        { 6.5 , 287.8852778150443 },
+        { 6.625 , 360.6709999914997 },
+        { 6.75 , 453.0107661026085 },
+        { 6.875 , 570.4128948692577 },
+        { 7.0 , 720.0 },
+        { 7.125 , 910.9984887224346 },
+        { 7.25 , 1155.38101391999 },
+        { 7.375 , 1468.710974602918 },
+        { 7.5 , 1871.254305797788 },
+        { 7.625 , 2389.445374943686 },
+        { 7.75 , 3057.822671192607 },
+        { 7.875 , 3921.588652226146 },
+        { 8.0 , 5040.0 },
+        { 8.125 , 6490.864232147346 },
+        { 8.25 , 8376.512350919926 },
+        { 8.375 , 10831.74343769652 },
+        { 8.5 , 14034.40729348341 },
+        { 8.625 , 18219.5209839456 },
+        { 8.75 , 23698.12570174271 },
+        { 8.875 , 30882.5106362809 },
+        { 9.0 , 40320.0 },
+        { 9.125 , 52738.27188619719 },
+        { 9.25 , 69106.22689508938 },
+        { 9.375 , 90715.85129070834 },
+        { 9.5 , 119292.461994609 },
+        { 9.625 , 157143.3684865308 },
+        { 9.75 , 207358.5998902487 },
+        { 9.875 , 274082.281896993 },
+        { 10.0 , 362880.0 },
+        { 10.125 , 481236.7309615494 },
+        { 10.25 , 639232.5987795768 },
+        { 10.375 , 850461.1058503906 },
+        { 10.5 , 1133278.388948786 },
+        { 10.625 , 1512504.921682859 },
+        { 10.75 , 2021746.348929925 },
+        { 10.875 , 2706562.533732806 },
+        { 11.0 , 3628800.0 },
+        { 11.125 , 4872521.900985687 },
+        { 11.25 , 6552134.137490662 },
+        { 11.375 , 8823533.973197803 },
+        { 11.5 , 1.1899423083962249e+7 },
+        { 11.625 , 1.6070364792880382e+7 },
+        { 11.75 , 2.1733773250996688e+7 },
+        { 11.875 , 2.943386755434427e+7 },
+        { 12.0 , 3.99168e+7 },
+        { 12.125 , 5.420680614846578e+7 },
+        { 12.25 , 7.371150904676994e+7 },
+        { 12.375 , 1.0036769894512501e+8 },
+        { 12.5 , 1.3684336546556586e+8 },
+        { 12.625 , 1.8681799071723443e+8 },
+        { 12.75 , 2.5537183569921107e+8 },
+        { 12.875 , 3.4952717720783816e+8 },
+        { 13.0 , 4.790016e+8 },
+        { 13.125 , 6.572575245501475e+8 },
+        { 13.25 , 9.029659858229319e+8 },
+        { 13.375 , 1.2420502744459219e+9 },
+        { 13.5 , 1.7105420683195732e+9 },
+        { 13.625 , 2.3585771328050845e+9 },
+        { 13.75 , 3.2559909051649416e+9 },
+        { 13.875 , 4.500162406550916e+9 },
+        { 14.0 , 6.2270208e+9 },
+        { 14.125 , 8.626505009720685e+9 },
+        { 14.25 , 1.196429931215385e+10 },
+        { 14.375 , 1.66124224207142e+10 },
+        { 14.5 , 2.309231792231424e+10 },
+        { 14.625 , 3.213561343446927e+10 },
+        { 14.75 , 4.476987494601794e+10 },
+        { 14.875 , 6.243975339089396e+10 },
+        { 15.0 , 8.71782912e+10 },
+        { 15.125 , 1.218493832623047e+11 },
+        { 15.25 , 1.704912651981923e+11 },
+        { 15.375 , 2.388035722977667e+11 },
+        { 15.5 , 3.348386098735565e+11 },
+        { 15.625 , 4.699833464791132e+11 },
+        { 15.75 , 6.603556554537646e+11 },
+        { 15.875 , 9.287913316895475e+11 },
+        { 16.0 , 1.307674368e+12 },
+        { 16.125 , 1.842971921842358e+12 },
+        { 16.25 , 2.599991794272433e+12 },
+        { 16.375 , 3.671604924078163e+12 },
+        { 16.5 , 5.189998453040126e+12 },
+        { 16.625 , 7.343489788736144e+12 },
+        { 16.75 , 1.040060157339679e+13 },
+        { 16.875 , 1.474456239057157e+13 },
+        { 17.0 , 2.0922789888e+13 },
+        { 17.125 , 2.971792223970803e+13 },
+        { 17.25 , 4.224986665692704e+13 },
+        { 17.375 , 6.012253063177992e+13 },
+        { 17.5 , 8.563497447516206e+13 },
+        { 17.625 , 1.220855177377384e+14 },
+        { 17.75 , 1.742100763543963e+14 },
+        { 17.875 , 2.488144903408952e+14 },
+        { 18.0 , 3.55687428096e+14 },
+        { 18.125 , 5.08919418355e+14 },
+        { 18.25 , 7.288101998319914e+14 },
+        { 18.375 , 1.044628969727176e+15 },
+        { 18.5 , 1.498612053315336e+15 },
+        { 18.625 , 2.151757250127639e+15 },
+        { 18.75 , 3.092228855290534e+15 },
+        { 18.875 , 4.447559014843502e+15 },
+        { 19.0 , 6.402373705728e+15 },
+        { 19.125 , 9.224164457684374e+15 },
+        { 19.25 , 1.330078614693384e+16 },
+        { 19.375 , 1.919505731873686e+16 },
+        { 19.5 , 2.772432298633372e+16 },
+        { 19.625 , 4.007647878362728e+16 },
+        { 19.75 , 5.797929103669752e+16 },
+        { 19.875 , 8.39476764051711e+16 },
+        { 20.0 , 1.21645100408832e+17 },
+        { 20.5 , 5.406242982335075e+17 },
+        { 21.0 , 2.43290200817664e+18 },
+        { 21.5 , 1.10827981137869e+19 },
+        { 22.0 , 5.109094217170944e+19 },
+        { 22.5 , 2.382801594464184e+20 },
+        { 23.0 , 1.124000727777608e+21 },
+        { 23.5 , 5.361303587544415e+21 },
+        { 24.0 , 2.585201673888498e+22 },
+        { 24.5 , 1.259906343072938e+23 },
+        { 25.0 , 6.204484017332395e+23 },
+        { 25.5 , 3.086770540528697e+24 },
+        { 26.0 , 1.551121004333099e+25 },
+        { 26.5 , 7.871264878348176e+25 },
+        { 27.0 , 4.032914611266056e+26 },
+        { 27.5 , 2.085885192762267e+27 },
+        { 28.0 , 1.088886945041835e+28 },
+        { 28.5 , 5.736184280096234e+28 },
+        { 29.0 , 3.048883446117139e+29 },
+        { 29.5 , 1.634812519827427e+30 },
+        { 30.0 , 8.841761993739702e+30 },
+        { 30.5 , 4.822696933490909e+31 },
+        { 31.0 , 2.65252859812191e+32 },
+        { 31.5 , 1.470922564714727e+33 },
+        { 32.0 , 8.222838654177922e+33 },
+        { 32.5 , 4.633406078851391e+34 },
+        { 33.0 , 2.631308369336935e+35 },
+        { 33.5 , 1.505856975626702e+36 },
+        { 34.0 , 8.683317618811885e+36 },
+        { 34.5 , 5.044620868349451e+37 },
+        { 35.0 , 2.952327990396041e+38 },
+        { 35.5 , 1.740394199580561e+39 },
+        { 36.0 , 1.033314796638614e+40 },
+        { 36.5 , 6.178399408510991e+40 },
+        { 37.0 , 3.719933267899013e+41 },
+        { 37.5 , 2.255115784106512e+42 },
+        { 38.0 , 1.376375309122634e+43 },
+        { 38.5 , 8.456684190399419e+43 },
+        { 39.0 , 5.230226174666011e+44 },
+        { 39.5 , 3.255823413303776e+45 },
+        { 40.0 , 2.039788208119745e+46 },
+        { 40.5 , 1.286050248254992e+47 },
+        { 41.0 , 8.159152832478976e+47 },
+        { 41.5 , 5.208503505432716e+48 },
+        { 42.0 , 3.345252661316381e+49 },
+        { 42.5 , 2.161528954754577e+50 },
+        { 43.0 , 1.40500611775288e+51 },
+        { 43.5 , 9.186498057706953e+51 },
+        { 44.0 , 6.041526306337383e+52 },
+        { 44.5 , 3.996126655102524e+53 },
+        { 45.0 , 2.658271574788449e+54 },
+        { 45.5 , 1.778276361520623e+55 },
+        { 46.0 , 1.196222208654802e+56 },
+        { 46.5 , 8.091157444918836e+56 },
+        { 47.0 , 5.502622159812088e+57 },
+        { 47.5 , 3.762388211887259e+58 },
+        { 48.0 , 2.586232415111682e+59 },
+        { 48.5 , 1.787134400646448e+60 },
+        { 49.0 , 1.241391559253607e+61 },
+        { 49.5 , 8.667601843135273e+61 },
+        { 50.0 , 6.082818640342675e+62 },
+        { 50.5 , 4.290462912351959e+63 },
+        { 51.0 , 3.041409320171338e+64 },
+        { 51.5 , 2.16668377073774e+65 },
+        { 52.0 , 1.551118753287382e+66 },
+        { 52.5 , 1.115842141929936e+67 },
+        { 53.0 , 8.065817517094388e+67 },
+        { 53.5 , 5.858171245132164e+68 },
+        { 54.0 , 4.274883284060025e+69 },
+        { 54.5 , 3.134121616145708e+70 },
+        { 55.0 , 2.308436973392413e+71 },
+        { 55.5 , 1.70809628079941e+72 },
+        { 56.0 , 1.269640335365828e+73 },
+        { 56.5 , 9.479934358436728e+73 },
+        { 57.0 , 7.109985878048635e+74 },
+        { 57.5 , 5.356162912516752e+75 },
+        { 58.0 , 4.052691950487721e+76 },
+        { 58.5 , 3.079793674697132e+77 },
+        { 59.0 , 2.350561331282878e+78 },
+        { 59.5 , 1.801679299697822e+79 },
+        { 60.0 , 1.386831185456898e+80 },
+        { 60.5 , 1.071999183320204e+81 },
+        { 61.0 , 8.320987112741391e+81 },
+        { 61.5 , 6.485595059087236e+82 },
+        { 62.0 , 5.075802138772249e+83 },
+        { 62.5 , 3.98864096133865e+84 },
+        { 63.0 , 3.146997326038794e+85 },
+        { 63.5 , 2.492900600836656e+86 },
+        { 64.0 , 1.98260831540444e+87 },
+        { 64.5 , 1.582991881531277e+88 },
+        { 65.0 , 1.268869321858841e+89 },
+        { 65.5 , 1.021029763587673e+90 },
+        { 66.0 , 8.247650592082472e+90 },
+        { 66.5 , 6.687744951499262e+91 },
+        { 67.0 , 5.443449390774431e+92 },
+        { 67.5 , 4.447350392747009e+93 },
+        { 68.0 , 3.647111091818868e+94 },
+        { 68.5 , 3.001961515104231e+95 },
+        { 69.0 , 2.48003554243683e+96 },
+        { 69.5 , 2.056343637846398e+97 },
+        { 70.0 , 1.711224524281413e+98 },
+        { 70.5 , 1.429158828303247e+99 },
+        { 71.0 , 1.19785716699699e+100 },
+        { 71.5 , 1.00755697395379e+101 },
+        { 72.0 , 8.50478588567862e+101 },
+        { 72.5 , 7.20403236376959e+102 },
+        { 73.0 , 6.12344583768861e+103 },
+        { 73.5 , 5.22292346373295e+104 },
+        { 74.0 , 4.47011546151268e+105 },
+        { 74.5 , 3.83884874584372e+106 },
+        { 75.0 , 3.30788544151939e+107 },
+        { 75.5 , 2.85994231565357e+108 },
+        { 76.0 , 2.48091408113954e+109 },
+        { 76.5 , 2.15925644831845e+110 },
+        { 77.0 , 1.88549470166605e+111 },
+        { 77.5 , 1.65183118296361e+112 },
+        { 78.0 , 1.45183092028286e+113 },
+        { 78.5 , 1.2801691667968e+114 },
+        { 79.0 , 1.13242811782063e+115 },
+        { 79.5 , 1.00493279593549e+116 },
+        { 80.0 , 8.94618213078298e+116 },
+        { 80.5 , 7.98921572768712e+117 },
+        { 81.0 , 7.15694570462638e+118 },
+        { 81.5 , 6.43131866078814e+119 },
+        { 82.0 , 5.79712602074737e+120 },
+        { 82.5 , 5.24152470854233e+121 },
+        { 83.0 , 4.75364333701284e+122 },
+        { 83.5 , 4.32425788454742e+123 },
+        { 84.0 , 3.94552396972066e+124 },
+        { 84.5 , 3.6107553335971e+125 },
+        { 85.0 , 3.31424013456535e+126 },
+        { 85.5 , 3.05108825688955e+127 },
+        { 86.0 , 2.81710411438055e+128 },
+        { 86.5 , 2.60868045964056e+129 },
+        { 87.0 , 2.42270953836727e+130 },
+        { 87.5 , 2.25650859758909e+131 },
+        { 88.0 , 2.10775729837953e+132 },
+        { 88.5 , 1.97444502289045e+133 },
+        { 89.0 , 1.85482642257398e+134 },
+        { 89.5 , 1.74738384525805e+135 },
+        { 90.0 , 1.65079551609085e+136 },
+        { 90.5 , 1.56390854150595e+137 },
+        { 91.0 , 1.48571596448176e+138 },
+        { 91.5 , 1.41533723006289e+139 },
+        { 92.0 , 1.3520015276784e+140 },
+        { 92.5 , 1.29503356550754e+141 },
+        { 93.0 , 1.24384140546413e+142 },
+        { 93.5 , 1.19790604809448e+143 },
+        { 94.0 , 1.15677250708164e+144 },
+        { 94.5 , 1.12004215496834e+145 },
+        { 95.0 , 1.08736615665674e+146 },
+        { 95.5 , 1.05843983644508e+147 },
+        { 96.0 , 1.03299784882391e+148 },
+        { 96.5 , 1.01081004380505e+149 },
+        { 97.0 , 9.9167793487095e+149 },
+        { 97.5 , 9.75431692271873e+150 },
+        { 98.0 , 9.61927596824821e+151 },
+        { 98.5 , 9.51045899965076e+152 },
+        { 99.0 , 9.42689044888325e+153 },
+        { 99.5 , 9.367802114656e+154 },
+        { 100.0 , 9.33262154439441e+155 },
+    };
+
+    @Test
+    public void testGamma() {
+
+        for (int i = 0; i < GAMMA_REF.length; i++) {
+            final double[] ref = GAMMA_REF[i];
+            final double x = ref[0];
+            final double expected = ref[1];
+            final double actual = Gamma.gamma(x);
+            final double absX = FastMath.abs(x);
+            final int ulps;
+            if (absX <= 8.0) {
+                ulps = 3;
+            } else if (absX <= 20.0) {
+                ulps = 5;
+            } else if (absX <= 30.0) {
+                ulps = 50;
+            } else if (absX <= 50.0) {
+                ulps = 180;
+            } else {
+                ulps = 500;
+            }
+            final double tol = ulps * FastMath.ulp(expected);
+            Assert.assertEquals(Double.toString(x), expected, actual, tol);
+        }
+    }
+
     private void checkRelativeError(String msg, double expected, double actual, double tolerance)
{
         Assert.assertEquals(msg, expected, actual, FastMath.abs(tolerance * actual));
     }



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