1/*- 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz. 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 * The argument reduction and testing for exceptional cases was 13 * written by Steven G. Kargl with input from Bruce D. Evans 14 * and David A. Schultz. 15 */ 16 17#include <float.h> 18#include <ieeefp.h> 19#include <math.h> 20 21#include "math_private.h" 22 23#define BIAS (LDBL_MAX_EXP - 1) 24 25static const unsigned 26 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ 27 28long double 29cbrtl(long double x) 30{ 31 long double v, r, s, t, w; 32 double dr, dt, dx; 33 float ft, fx; 34 uint64_t hx, lx; 35 uint16_t expsign; 36 int k; 37 38 GET_LDOUBLE_MSW64(hx,x); 39 k = (hx>>48)&0x7fff; 40 41 /* 42 * If x = +-Inf, then cbrt(x) = +-Inf. 43 * If x = NaN, then cbrt(x) = NaN. 44 */ 45 if (k == BIAS + LDBL_MAX_EXP) 46 return (x + x); 47 48 if (k == 0) { 49 /* If x = +-0, then cbrt(x) = +-0. */ 50 GET_LDOUBLE_WORDS64(hx,lx,x); 51 if (((hx&0x7fffffffffffffffLL)|lx) == 0) { 52 return (x); 53 } 54 /* Adjust subnormal numbers. */ 55 x *= 0x1.0p514; 56 GET_LDOUBLE_MSW64(hx,x); 57 k = (hx>>48)&0x7fff; 58 k -= BIAS + 514; 59 } else 60 k -= BIAS; 61 GET_LDOUBLE_MSW64(hx,x); 62 hx = (hx&0x8000ffffffffffffLL)|((uint64_t)BIAS<<48); 63 SET_LDOUBLE_MSW64(x,hx); 64 v = 1; 65 66 switch (k % 3) { 67 case 1: 68 case -2: 69 x = 2*x; 70 k--; 71 break; 72 case 2: 73 case -1: 74 x = 4*x; 75 k -= 2; 76 break; 77 } 78 GET_LDOUBLE_MSW64(hx,x); 79 expsign = ((hx>>48) & 0x8000) | (BIAS + k / 3); 80 hx = (hx&0x8000ffffffffffffLL)|((uint64_t)expsign<<48); 81 SET_LDOUBLE_MSW64(x,hx); 82 83 /* 84 * The following is the guts of s_cbrtf, with the handling of 85 * special values removed and extra care for accuracy not taken, 86 * but with most of the extra accuracy not discarded. 87 */ 88 89 /* ~5-bit estimate: */ 90 fx = x; 91 GET_FLOAT_WORD(hx, fx); 92 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1)); 93 94 /* ~16-bit estimate: */ 95 dx = x; 96 dt = ft; 97 dr = dt * dt * dt; 98 dt = dt * (dx + dx + dr) / (dx + dr + dr); 99 100 /* ~47-bit estimate: */ 101 dr = dt * dt * dt; 102 dt = dt * (dx + dx + dr) / (dx + dr + dr); 103 104 /* 105 * Round dt away from zero to 47 bits. Since we don't trust the 47, 106 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and 107 * might be avoidable in this case, since on most machines dt will 108 * have been evaluated in 53-bit precision and the technical reasons 109 * for rounding up might not apply to either case in cbrtl() since 110 * dt is much more accurate than needed. 111 */ 112 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60; 113 114 /* 115 * Final step Newton iteration to 64 or 113 bits with 116 * error < 0.667 ulps 117 */ 118 s=t*t; /* t*t is exact */ 119 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 120 w=t+t; /* t+t is exact */ 121 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 122 t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ 123 124 t *= v; 125 return (t); 126} 127