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 #ifdef __i386__ 19 #include <ieeefp.h> 20 #endif 21 22 #include "fpmath.h" 23 #include "math.h" 24 #include "math_private.h" 25 26 #define BIAS (LDBL_MAX_EXP - 1) 27 28 static const unsigned 29 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ 30 31 long double 32 cbrtl(long double x) 33 { 34 union IEEEl2bits u, v; 35 long double r, s, t, w; 36 double dr, dt, dx; 37 float ft, fx; 38 uint32_t hx; 39 uint16_t expsign; 40 int k; 41 42 u.e = x; 43 expsign = u.xbits.expsign; 44 k = expsign & 0x7fff; 45 46 /* 47 * If x = +-Inf, then cbrt(x) = +-Inf. 48 * If x = NaN, then cbrt(x) = NaN. 49 */ 50 if (k == BIAS + LDBL_MAX_EXP) 51 return (x + x); 52 53 ENTERI(); 54 if (k == 0) { 55 /* If x = +-0, then cbrt(x) = +-0. */ 56 if ((u.bits.manh | u.bits.manl) == 0) 57 RETURNI(x); 58 /* Adjust subnormal numbers. */ 59 u.e *= 0x1.0p514; 60 k = u.bits.exp; 61 k -= BIAS + 514; 62 } else 63 k -= BIAS; 64 u.xbits.expsign = BIAS; 65 v.e = 1; 66 67 x = u.e; 68 switch (k % 3) { 69 case 1: 70 case -2: 71 x = 2*x; 72 k--; 73 break; 74 case 2: 75 case -1: 76 x = 4*x; 77 k -= 2; 78 break; 79 } 80 v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3); 81 82 /* 83 * The following is the guts of s_cbrtf, with the handling of 84 * special values removed and extra care for accuracy not taken, 85 * but with most of the extra accuracy not discarded. 86 */ 87 88 /* ~5-bit estimate: */ 89 fx = x; 90 GET_FLOAT_WORD(hx, fx); 91 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1)); 92 93 /* ~16-bit estimate: */ 94 dx = x; 95 dt = ft; 96 dr = dt * dt * dt; 97 dt = dt * (dx + dx + dr) / (dx + dr + dr); 98 99 /* ~47-bit estimate: */ 100 dr = dt * dt * dt; 101 dt = dt * (dx + dx + dr) / (dx + dr + dr); 102 103 #if LDBL_MANT_DIG == 64 104 /* 105 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8). 106 * Round it away from zero to 32 bits (32 so that t*t is exact, and 107 * away from zero for technical reasons). 108 */ 109 volatile double vd2 = 0x1.0p32; 110 volatile double vd1 = 0x1.0p-31; 111 #define vd ((long double)vd2 + vd1) 112 113 t = dt + vd - 0x1.0p32; 114 #elif LDBL_MANT_DIG == 113 115 /* 116 * Round dt away from zero to 47 bits. Since we don't trust the 47, 117 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and 118 * might be avoidable in this case, since on most machines dt will 119 * have been evaluated in 53-bit precision and the technical reasons 120 * for rounding up might not apply to either case in cbrtl() since 121 * dt is much more accurate than needed. 122 */ 123 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60; 124 #else 125 #error "Unsupported long double format" 126 #endif 127 128 /* 129 * Final step Newton iteration to 64 or 113 bits with 130 * error < 0.667 ulps 131 */ 132 s=t*t; /* t*t is exact */ 133 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 134 w=t+t; /* t+t is exact */ 135 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 136 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ 137 138 t *= v.e; 139 RETURNI(t); 140 } 141