1 /* @(#)s_cbrt.c 5.1 93/09/24 */ 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 15 #include <sys/cdefs.h> 16 __FBSDID("$FreeBSD$"); 17 18 #include "math.h" 19 #include "math_private.h" 20 21 /* cbrt(x) 22 * Return cube root of x 23 */ 24 static const u_int32_t 25 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 26 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 27 28 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ 29 static const double 30 P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ 31 P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ 32 P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ 33 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ 34 P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ 35 36 double 37 cbrt(double x) 38 { 39 int32_t hx; 40 union { 41 double value; 42 uint64_t bits; 43 } u; 44 double r,s,t=0.0,w; 45 u_int32_t sign; 46 u_int32_t high,low; 47 48 EXTRACT_WORDS(hx,low,x); 49 sign=hx&0x80000000; /* sign= sign(x) */ 50 hx ^=sign; 51 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 52 53 /* 54 * Rough cbrt to 5 bits: 55 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 56 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 57 * "%" are integer division and modulus with rounding towards minus 58 * infinity. The RHS is always >= the LHS and has a maximum relative 59 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 60 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 61 * floating point representation, for finite positive normal values, 62 * ordinary integer divison of the value in bits magically gives 63 * almost exactly the RHS of the above provided we first subtract the 64 * exponent bias (1023 for doubles) and later add it back. We do the 65 * subtraction virtually to keep e >= 0 so that ordinary integer 66 * division rounds towards minus infinity; this is also efficient. 67 */ 68 if(hx<0x00100000) { /* zero or subnormal? */ 69 if((hx|low)==0) 70 return(x); /* cbrt(0) is itself */ 71 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 72 t*=x; 73 GET_HIGH_WORD(high,t); 74 INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0); 75 } else 76 INSERT_WORDS(t,sign|(hx/3+B1),0); 77 78 /* 79 * New cbrt to 23 bits: 80 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) 81 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) 82 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation 83 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this 84 * gives us bounds for r = t**3/x. 85 * 86 * Try to optimize for parallel evaluation as in k_tanf.c. 87 */ 88 r=(t*t)*(t/x); 89 t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); 90 91 /* 92 * Round t away from zero to 23 bits (sloppily except for ensuring that 93 * the result is larger in magnitude than cbrt(x) but not much more than 94 * 2 23-bit ulps larger). With rounding towards zero, the error bound 95 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps 96 * in the rounded t, the infinite-precision error in the Newton 97 * approximation barely affects third digit in the the final error 98 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps 99 * before the final error is larger than 0.667 ulps. 100 */ 101 u.value=t; 102 u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL; 103 t=u.value; 104 105 /* one step Newton iteration to 53 bits with error < 0.667 ulps */ 106 s=t*t; /* t*t is exact */ 107 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 108 w=t+t; /* t+t is exact */ 109 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 110 t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ 111 112 return(t); 113 } 114