compiler/stdc/math/s_cbrt.c

   1 /* @(#)s_cbrt.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  *
  12  * Optimized by Bruce D. Evans.
  13  */
  14
  15 #ifndef lint
  16 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.17 2011/03/12 16:50:39 kargl Exp $";
  17 #endif
  18
  19 #include <float.h>
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 /* cbrt(x)
  24  * Return cube root of x
  25  */
  26 static const uint32_t
  27         B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
  28         B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
  29
  30 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
  31 static const double
  32 P0 =  1.87595182427177009643,           /* 0x3ffe03e6, 0x0f61e692 */
  33 P1 = -1.88497979543377169875,           /* 0xbffe28e0, 0x92f02420 */
  34 P2 =  1.621429720105354466140,          /* 0x3ff9f160, 0x4a49d6c2 */
  35 P3 = -0.758397934778766047437,          /* 0xbfe844cb, 0xbee751d9 */
  36 P4 =  0.145996192886612446982;          /* 0x3fc2b000, 0xd4e4edd7 */
  37
  38 double
  39 cbrt(double x)
  40 {
  41         int32_t hx;
  42         union {
  43             double value;
  44             uint64_t bits;
  45         } u;
  46         double r,s,t=0.0,w;
  47         uint32_t sign;
  48         uint32_t high,low;
  49
  50         EXTRACT_WORDS(hx,low,x);
  51         sign=hx&0x80000000;             /* sign= sign(x) */
  52         hx  ^=sign;
  53         if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
  54
  55     /*
  56      * Rough cbrt to 5 bits:
  57      *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
  58      * where e is integral and >= 0, m is real and in [0, 1), and "/" and
  59      * "%" are integer division and modulus with rounding towards minus
  60      * infinity.  The RHS is always >= the LHS and has a maximum relative
  61      * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
  62      * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
  63      * floating point representation, for finite positive normal values,
  64      * ordinary integer division of the value in bits magically gives
  65      * almost exactly the RHS of the above provided we first subtract the
  66      * exponent bias (1023 for doubles) and later add it back.  We do the
  67      * subtraction virtually to keep e >= 0 so that ordinary integer
  68      * division rounds towards minus infinity; this is also efficient.
  69      */
  70         if(hx<0x00100000) {             /* zero or subnormal? */
  71             if((hx|low)==0)
  72                 return(x);              /* cbrt(0) is itself */
  73             SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
  74             t*=x;
  75             GET_HIGH_WORD(high,t);
  76             INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
  77         } else
  78             INSERT_WORDS(t,sign|(hx/3+B1),0);
  79
  80     /*
  81      * New cbrt to 23 bits:
  82      *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
  83      * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
  84      * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
  85      * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
  86      * gives us bounds for r = t**3/x.
  87      *
  88      * Try to optimize for parallel evaluation as in k_tanf.c.
  89      */
  90         r=(t*t)*(t/x);
  91         t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
  92
  93     /*
  94      * Round t away from zero to 23 bits (sloppily except for ensuring that
  95      * the result is larger in magnitude than cbrt(x) but not much more than
  96      * 2 23-bit ulps larger).  With rounding towards zero, the error bound
  97      * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
  98      * in the rounded t, the infinite-precision error in the Newton
  99      * approximation barely affects third digit in the final error
 100      * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
 101      * before the final error is larger than 0.667 ulps.
 102      */
 103         u.value=t;
 104         u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
 105         t=u.value;
 106
 107     /* one step Newton iteration to 53 bits with error < 0.667 ulps */
 108         s=t*t;                          /* t*t is exact */
 109         r=x/s;                          /* error <= 0.5 ulps; |r| < |t| */
 110         w=t+t;                          /* t+t is exact */
 111         r=(r-t)/(w+r);                  /* r-t is exact; w+r ~= 3*t */
 112         t=t+t*r;                        /* error <= 0.5 + 0.5/3 + epsilon */
 113
 114         return(t);
 115 }
 116
 117 #if     LDBL_MANT_DIG == DBL_MANT_DIG
 118 AROS_MAKE_ASM_SYM(typeof(cbrtl), cbrtl, AROS_CSYM_FROM_ASM_NAME(cbrtl), AROS_CSYM_FROM_ASM_NAME(cbrt));
 119 AROS_EXPORT_ASM_SYM(AROS_CSYM_FROM_ASM_NAME(cbrtl));
 120 #endif