gonzui

    1: /*      $OpenBSD: b_tgamma.c,v 1.4 2011/07/06 00:02:42 martynas Exp $        */
    2: /*-
    3:  * Copyright (c) 1992, 1993
    4:  *      The Regents of the University of California.  All rights reserved.
    5:  *
    6:  * Redistribution and use in source and binary forms, with or without
    7:  * modification, are permitted provided that the following conditions
    8:  * are met:
    9:  * 1. Redistributions of source code must retain the above copyright
   10:  *    notice, this list of conditions and the following disclaimer.
   11:  * 2. Redistributions in binary form must reproduce the above copyright
   12:  *    notice, this list of conditions and the following disclaimer in the
   13:  *    documentation and/or other materials provided with the distribution.
   14:  * 3. Neither the name of the University nor the names of its contributors
   15:  *    may be used to endorse or promote products derived from this software
   16:  *    without specific prior written permission.
   17:  *
   18:  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
   19:  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   20:  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
   21:  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
   22:  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
   23:  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
   24:  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
   25:  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
   26:  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
   27:  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
   28:  * SUCH DAMAGE.
   29:  */
   30: 
   31: /* LINTLIBRARY */
   32: 
   33: /*
   34:  * This code by P. McIlroy, Oct 1992;
   35:  *
   36:  * The financial support of UUNET Communications Services is greatfully
   37:  * acknowledged.
   38:  */
   39: 
   40: #include <sys/cdefs.h>
   41: #include <float.h>
   42: #include <math.h>
   43: 
   44: #include "math_private.h"
   45: 
   46: /* METHOD:
   47:  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
   48:  *      At negative integers, return NaN and raise invalid.
   49:  *
   50:  * x < 6.5:
   51:  *      Use argument reduction G(x+1) = xG(x) to reach the
   52:  *      range [1.066124,2.066124].  Use a rational
   53:  *      approximation centered at the minimum (x0+1) to
   54:  *      ensure monotonicity.
   55:  *
   56:  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
   57:  *      adjusted for equal-ripples:
   58:  *
   59:  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
   60:  *
   61:  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
   62:  *      avoid premature round-off.
   63:  *
   64:  * Special values:
   65:  *      -Inf:                        return NaN and raise invalid;
   66:  *      negative integer:    return NaN and raise invalid;
   67:  *      other x ~< -177.79:  return +-0 and raise underflow;
   68:  *      +-0:                 return +-Inf and raise divide-by-zero;
   69:  *      finite x ~> 171.63:  return +Inf and raise overflow;
   70:  *      +Inf:                        return +Inf;
   71:  *      NaN:                         return NaN.
   72:  *
   73:  * Accuracy: tgamma(x) is accurate to within
   74:  *      x > 0:  error provably < 0.9ulp.
   75:  *      Maximum observed in 1,000,000 trials was .87ulp.
   76:  *      x < 0:
   77:  *      Maximum observed error < 4ulp in 1,000,000 trials.
   78:  */
   79: 
   80: static double neg_gam(double);
   81: static double small_gam(double);
   82: static double smaller_gam(double);
   83: static struct Double large_gam(double);
   84: static struct Double ratfun_gam(double, double);
   85: 
   86: /*
   87:  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
   88:  * [1.066.., 2.066..] accurate to 4.25e-19.
   89:  */
   90: #define LEFT -.3955078125       /* left boundary for rat. approx */
   91: #define x0 .461632144968362356785       /* xmin - 1 */
   92: 
   93: #define a0_hi 0.88560319441088874992
   94: #define a0_lo -.00000000000000004996427036469019695
   95: #define P0       6.21389571821820863029017800727e-01
   96: #define P1       2.65757198651533466104979197553e-01
   97: #define P2       5.53859446429917461063308081748e-03
   98: #define P3       1.38456698304096573887145282811e-03
   99: #define P4       2.40659950032711365819348969808e-03
  100: #define Q0       1.45019531250000000000000000000e+00
  101: #define Q1       1.06258521948016171343454061571e+00
  102: #define Q2      -2.07474561943859936441469926649e-01
  103: #define Q3      -1.46734131782005422506287573015e-01
  104: #define Q4       3.07878176156175520361557573779e-02
  105: #define Q5       5.12449347980666221336054633184e-03
  106: #define Q6      -1.76012741431666995019222898833e-03
  107: #define Q7       9.35021023573788935372153030556e-05
  108: #define Q8       6.13275507472443958924745652239e-06
  109: /*
  110:  * Constants for large x approximation (x in [6, Inf])
  111:  * (Accurate to 2.8*10^-19 absolute)
  112:  */
  113: #define lns2pi_hi 0.418945312500000
  114: #define lns2pi_lo -.000006779295327258219670263595
  115: #define Pa0      8.33333333333333148296162562474e-02
  116: #define Pa1     -2.77777777774548123579378966497e-03
  117: #define Pa2      7.93650778754435631476282786423e-04
  118: #define Pa3     -5.95235082566672847950717262222e-04
  119: #define Pa4      8.41428560346653702135821806252e-04
  120: #define Pa5     -1.89773526463879200348872089421e-03
  121: #define Pa6      5.69394463439411649408050664078e-03
  122: #define Pa7     -1.44705562421428915453880392761e-02
  123: 
  124: static const double zero = 0., one = 1.0, tiny = 1e-300;
  125: 
  126: double
  127: tgamma(double x)
  128: {
  129:         struct Double u;
  130: 
  131:         if (x >= 6) {
  132:                 if(x > 171.63)
  133:                         return(x/zero);
  134:                 u = large_gam(x);
  135:                 return(__exp__D(u.a, u.b));
  136:         } else if (x >= 1.0 + LEFT + x0)
  137:                 return (small_gam(x));
  138:         else if (x > 1.e-17)
  139:                 return (smaller_gam(x));
  140:         else if (x > -1.e-17) {
  141:                 if (x != 0.0)
  142:                         u.a = one - tiny;    /* raise inexact */
  143:                 return (one/x);
  144:         } else if (!finite(x)) {
  145:                 return (x - x);                       /* x = NaN, -Inf */
  146:          } else
  147:                 return (neg_gam(x));
  148: }
  149: 
  150: /*
  151:  * We simply call tgamma() rather than bloating the math library
  152:  * with a float-optimized version of it.  The reason is that tgammaf()
  153:  * is essentially useless, since the function is superexponential
  154:  * and floats have very limited range.  -- das@freebsd.org
  155:  */
  156: 
  157: float
  158: tgammaf(float x)
  159: {
  160:         return tgamma(x);
  161: }
  162: 
  163: /*
  164:  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
  165:  */
  166: 
  167: static struct Double
  168: large_gam(double x)
  169: {
  170:         double z, p;
  171:         struct Double t, u, v;
  172: 
  173:         z = one/(x*x);
  174:         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
  175:         p = p/x;
  176: 
  177:         u = __log__D(x);
  178:         u.a -= one;
  179:         v.a = (x -= .5);
  180:         TRUNC(v.a);
  181:         v.b = x - v.a;
  182:         t.a = v.a*u.a;                 /* t = (x-.5)*(log(x)-1) */
  183:         t.b = v.b*u.a + x*u.b;
  184:         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
  185:         t.b += lns2pi_lo; t.b += p;
  186:         u.a = lns2pi_hi + t.b; u.a += t.a;
  187:         u.b = t.a - u.a;
  188:         u.b += lns2pi_hi; u.b += t.b;
  189:         return (u);
  190: }
  191: 
  192: /*
  193:  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
  194:  * It also has correct monotonicity.
  195:  */
  196: 
  197: static double
  198: small_gam(double x)
  199: {
  200:         double y, ym1, t;
  201:         struct Double yy, r;
  202:         y = x - one;
  203:         ym1 = y - one;
  204:         if (y <= 1.0 + (LEFT + x0)) {
  205:                 yy = ratfun_gam(y - x0, 0);
  206:                 return (yy.a + yy.b);
  207:         }
  208:         r.a = y;
  209:         TRUNC(r.a);
  210:         yy.a = r.a - one;
  211:         y = ym1;
  212:         yy.b = r.b = y - yy.a;
  213:         /* Argument reduction: G(x+1) = x*G(x) */
  214:         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
  215:                 t = r.a*yy.a;
  216:                 r.b = r.a*yy.b + y*r.b;
  217:                 r.a = t;
  218:                 TRUNC(r.a);
  219:                 r.b += (t - r.a);
  220:         }
  221:         /* Return r*tgamma(y). */
  222:         yy = ratfun_gam(y - x0, 0);
  223:         y = r.b*(yy.a + yy.b) + r.a*yy.b;
  224:         y += yy.a*r.a;
  225:         return (y);
  226: }
  227: 
  228: /*
  229:  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
  230:  */
  231: 
  232: static double
  233: smaller_gam(double x)
  234: {
  235:         double t, d;
  236:         struct Double r, xx;
  237:         if (x < x0 + LEFT) {
  238:                 t = x;
  239:                 TRUNC(t);
  240:                 d = (t+x)*(x-t);
  241:                 t *= t;
  242:                 xx.a = (t + x);
  243:                 TRUNC(xx.a);
  244:                 xx.b = x - xx.a; xx.b += t; xx.b += d;
  245:                 t = (one-x0); t += x;
  246:                 d = (one-x0); d -= t; d += x;
  247:                 x = xx.a + xx.b;
  248:         } else {
  249:                 xx.a =  x;
  250:                 TRUNC(xx.a);
  251:                 xx.b = x - xx.a;
  252:                 t = x - x0;
  253:                 d = (-x0 -t); d += x;
  254:         }
  255:         r = ratfun_gam(t, d);
  256:         d = r.a/x;
  257:         TRUNC(d);
  258:         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
  259:         return (d + r.a/x);
  260: }
  261: 
  262: /*
  263:  * returns (z+c)^2 * P(z)/Q(z) + a0
  264:  */
  265: 
  266: static struct Double
  267: ratfun_gam(double z, double c)
  268: {
  269:         double p, q;
  270:         struct Double r, t;
  271: 
  272:         q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
  273:         p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
  274: 
  275:         /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
  276:         p = p/q;
  277:         t.a = z;
  278:         TRUNC(t.a);                    /* t ~= z + c */
  279:         t.b = (z - t.a) + c;
  280:         t.b *= (t.a + z);
  281:         q = (t.a *= t.a);              /* t = (z+c)^2 */
  282:         TRUNC(t.a);
  283:         t.b += (q - t.a);
  284:         r.a = p;
  285:         TRUNC(r.a);                    /* r = P/Q */
  286:         r.b = p - r.a;
  287:         t.b = t.b*p + t.a*r.b + a0_lo;
  288:         t.a *= r.a;                    /* t = (z+c)^2*(P/Q) */
  289:         r.a = t.a + a0_hi;
  290:         TRUNC(r.a);
  291:         r.b = ((a0_hi-r.a) + t.a) + t.b;
  292:         return (r);                    /* r = a0 + t */
  293: }
  294: 
  295: static double
  296: neg_gam(double x)
  297: {
  298:         int sgn = 1;
  299:         struct Double lg, lsine;
  300:         double y, z;
  301: 
  302:         y = ceil(x);
  303:         if (y == x)            /* Negative integer. */
  304:                 return ((x - x) / zero);
  305:         z = y - x;
  306:         if (z > 0.5)
  307:                 z = one - z;
  308:         y = 0.5 * y;
  309:         if (y == ceil(y))
  310:                 sgn = -1;
  311:         if (z < .25)
  312:                 z = sin(M_PI*z);
  313:         else
  314:                 z = cos(M_PI*(0.5-z));
  315:         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
  316:         if (x < -170) {
  317:                 if (x < -190)
  318:                         return ((double)sgn*tiny*tiny);
  319:                 y = one - x;          /* exact: 128 < |x| < 255 */
  320:                 lg = large_gam(y);
  321:                 lsine = __log__D(M_PI/z);     /* = TRUNC(log(u)) + small */
  322:                 lg.a -= lsine.a;              /* exact (opposite signs) */
  323:                 lg.b -= lsine.b;
  324:                 y = -(lg.a + lg.b);
  325:                 z = (y + lg.a) + lg.b;
  326:                 y = __exp__D(y, z);
  327:                 if (sgn < 0) y = -y;
  328:                 return (y);
  329:         }
  330:         y = one-x;
  331:         if (one-y == x)
  332:                 y = tgamma(y);
  333:         else           /* 1-x is inexact */
  334:                 y = -x*tgamma(-x);
  335:         if (sgn < 0) y = -y;
  336:         return (M_PI / (y*z));
  337: }
  338: 
  339: #if     LDBL_MANT_DIG == 53
  340: #ifdef  lint
  341: /* PROTOLIB1 */
  342: long double tgammal(long double);
  343: #else   /* lint */
  344: __weak_alias(tgammal, tgamma);
  345: #endif  /* lint */
  346: #endif  /* LDBL_MANT_DIG == 53 */