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#include <sys/cdefs.h> 17__RCSID("$NetBSD: s_cbrtl.c,v 1.4 2024/04/03 18:53:42 christos Exp $"); 18 19 20#include "namespace.h" 21#include <machine/ieee.h> 22#include <float.h> 23 24#include "math.h" 25#include "math_private.h" 26 27#ifdef __HAVE_LONG_DOUBLE 28__weak_alias(cbrtl, _cbrtl) 29 30#define BIAS (LDBL_MAX_EXP - 1) 31 32static const unsigned 33 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ 34 35long double 36cbrtl(long double x) 37{ 38 union ieee_ext_u u, v; 39 long double r, s, t, w; 40 double dr, dt, dx; 41 float ft, fx; 42 uint32_t hx; 43 uint16_t expsign; 44 int k; 45 46 u.extu_ld = x; 47 expsign = GET_EXPSIGN(&u); 48 k = expsign & 0x7fff; 49 50 /* 51 * If x = +-Inf, then cbrt(x) = +-Inf. 52 * If x = NaN, then cbrt(x) = NaN. 53 */ 54 if (k == BIAS + LDBL_MAX_EXP) 55 return (x + x); 56 57 ENTERI(); 58 if (k == 0) { 59 /* If x = +-0, then cbrt(x) = +-0. */ 60 if ((u.extu_frach | u.extu_fracl) == 0) 61 RETURNI(x); 62 /* Adjust subnormal numbers. */ 63 u.extu_ld *= 0x1.0p514; 64 k = u.extu_exp; 65 k -= BIAS + 514; 66 } else 67 k -= BIAS; 68 SET_EXPSIGN(&u, BIAS); 69 v.extu_ld = 1; 70 71 x = u.extu_ld; 72 switch (k % 3) { 73 case 1: 74 case -2: 75 x = 2*x; 76 k--; 77 break; 78 case 2: 79 case -1: 80 x = 4*x; 81 k -= 2; 82 break; 83 } 84 SET_EXPSIGN(&v, (expsign & 0x8000) | (BIAS + k / 3)); 85 86 /* 87 * The following is the guts of s_cbrtf, with the handling of 88 * special values removed and extra care for accuracy not taken, 89 * but with most of the extra accuracy not discarded. 90 */ 91 92 /* ~5-bit estimate: */ 93 fx = x; 94 GET_FLOAT_WORD(hx, fx); 95 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1)); 96 97 /* ~16-bit estimate: */ 98 dx = x; 99 dt = ft; 100 dr = dt * dt * dt; 101 dt = dt * (dx + dx + dr) / (dx + dr + dr); 102 103 /* ~47-bit estimate: */ 104 dr = dt * dt * dt; 105 dt = dt * (dx + dx + dr) / (dx + dr + dr); 106 107#if LDBL_MANT_DIG == 64 108 /* 109 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8). 110 * Round it away from zero to 32 bits (32 so that t*t is exact, and 111 * away from zero for technical reasons). 112 */ 113 volatile double vd2 = 0x1.0p32; 114 volatile double vd1 = 0x1.0p-31; 115 #define vd ((long double)vd2 + vd1) 116 117 t = dt + vd - 0x1.0p32; 118#elif LDBL_MANT_DIG == 113 119 /* 120 * Round dt away from zero to 47 bits. Since we don't trust the 47, 121 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and 122 * might be avoidable in this case, since on most machines dt will 123 * have been evaluated in 53-bit precision and the technical reasons 124 * for rounding up might not apply to either case in cbrtl() since 125 * dt is much more accurate than needed. 126 */ 127 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60; 128#else 129#error "Unsupported long double format" 130#endif 131 132 /* 133 * Final step Newton iteration to 64 or 113 bits with 134 * error < 0.667 ulps 135 */ 136 s=t*t; /* t*t is exact */ 137 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 138 w=t+t; /* t+t is exact */ 139 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 140 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ 141 142 t *= v.extu_ld; 143 RETURNI(t); 144} 145#endif /* __HAVE_LONG_DOUBLE */ 146