1 /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */ 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 /* cbrt(x) 15 * Return cube root of x 16 */ 17 18 #include <math.h> 19 #include <stdint.h> 20 21 static const uint32_t 22 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 23 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 24 25 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ 26 static const double 27 P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ 28 P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ 29 P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ 30 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ 31 P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ 32 33 double cbrt(double x) 34 { 35 union {double f; uint64_t i;} u = {x}; 36 double_t r,s,t,w; 37 uint32_t hx = u.i>>32 & 0x7fffffff; 38 39 if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */ 40 return x+x; 41 42 /* 43 * Rough cbrt to 5 bits: 44 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 45 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 46 * "%" are integer division and modulus with rounding towards minus 47 * infinity. The RHS is always >= the LHS and has a maximum relative 48 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 49 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 50 * floating point representation, for finite positive normal values, 51 * ordinary integer divison of the value in bits magically gives 52 * almost exactly the RHS of the above provided we first subtract the 53 * exponent bias (1023 for doubles) and later add it back. We do the 54 * subtraction virtually to keep e >= 0 so that ordinary integer 55 * division rounds towards minus infinity; this is also efficient. 56 */ 57 if (hx < 0x00100000) { /* zero or subnormal? */ 58 u.f = x*0x1p54; 59 hx = u.i>>32 & 0x7fffffff; 60 if (hx == 0) 61 return x; /* cbrt(0) is itself */ 62 hx = hx/3 + B2; 63 } else 64 hx = hx/3 + B1; 65 u.i &= 1ULL<<63; 66 u.i |= (uint64_t)hx << 32; 67 t = u.f; 68 69 /* 70 * New cbrt to 23 bits: 71 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) 72 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) 73 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation 74 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this 75 * gives us bounds for r = t**3/x. 76 * 77 * Try to optimize for parallel evaluation as in __tanf.c. 78 */ 79 r = (t*t)*(t/x); 80 t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); 81 82 /* 83 * Round t away from zero to 23 bits (sloppily except for ensuring that 84 * the result is larger in magnitude than cbrt(x) but not much more than 85 * 2 23-bit ulps larger). With rounding towards zero, the error bound 86 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps 87 * in the rounded t, the infinite-precision error in the Newton 88 * approximation barely affects third digit in the final error 89 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps 90 * before the final error is larger than 0.667 ulps. 91 */ 92 u.f = t; 93 u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL; 94 t = u.f; 95 96 /* one step Newton iteration to 53 bits with error < 0.667 ulps */ 97 s = t*t; /* t*t is exact */ 98 r = x/s; /* error <= 0.5 ulps; |r| < |t| */ 99 w = t+t; /* t+t is exact */ 100 r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 101 t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ 102 return t; 103 } 104