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 <sys/cdefs.h> 18 #include <float.h> 19 #ifdef __i386__ 20 #include <ieeefp.h> 21 #endif 22 23 #include "fpmath.h" 24 #include "math.h" 25 #include "math_private.h" 26 27 #define BIAS (LDBL_MAX_EXP - 1) 28 29 static const unsigned 30 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ 31 32 long double 33 cbrtl(long double x) 34 { 35 union IEEEl2bits u, v; 36 long double r, s, t, w; 37 double dr, dt, dx; 38 float ft, fx; 39 uint32_t hx; 40 uint16_t expsign; 41 int k; 42 43 u.e = x; 44 expsign = u.xbits.expsign; 45 k = expsign & 0x7fff; 46 47 /* 48 * If x = +-Inf, then cbrt(x) = +-Inf. 49 * If x = NaN, then cbrt(x) = NaN. 50 */ 51 if (k == BIAS + LDBL_MAX_EXP) 52 return (x + x); 53 54 ENTERI(); 55 if (k == 0) { 56 /* If x = +-0, then cbrt(x) = +-0. */ 57 if ((u.bits.manh | u.bits.manl) == 0) 58 RETURNI(x); 59 /* Adjust subnormal numbers. */ 60 u.e *= 0x1.0p514; 61 k = u.bits.exp; 62 k -= BIAS + 514; 63 } else 64 k -= BIAS; 65 u.xbits.expsign = BIAS; 66 v.e = 1; 67 68 x = u.e; 69 switch (k % 3) { 70 case 1: 71 case -2: 72 x = 2*x; 73 k--; 74 break; 75 case 2: 76 case -1: 77 x = 4*x; 78 k -= 2; 79 break; 80 } 81 v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3); 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 #if LDBL_MANT_DIG == 64 105 /* 106 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8). 107 * Round it away from zero to 32 bits (32 so that t*t is exact, and 108 * away from zero for technical reasons). 109 */ 110 volatile double vd2 = 0x1.0p32; 111 volatile double vd1 = 0x1.0p-31; 112 #define vd ((long double)vd2 + vd1) 113 114 t = dt + vd - 0x1.0p32; 115 #elif LDBL_MANT_DIG == 113 116 /* 117 * Round dt away from zero to 47 bits. Since we don't trust the 47, 118 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and 119 * might be avoidable in this case, since on most machines dt will 120 * have been evaluated in 53-bit precision and the technical reasons 121 * for rounding up might not apply to either case in cbrtl() since 122 * dt is much more accurate than needed. 123 */ 124 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60; 125 #else 126 #error "Unsupported long double format" 127 #endif 128 129 /* 130 * Final step Newton iteration to 64 or 113 bits with 131 * error < 0.667 ulps 132 */ 133 s=t*t; /* t*t is exact */ 134 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 135 w=t+t; /* t+t is exact */ 136 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 137 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ 138 139 t *= v.e; 140 RETURNI(t); 141 } 142