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 #include <float.h> 17 #include "math.h" 18 #include "math_private.h" 19 20 /* cbrt(x) 21 * Return cube root of x 22 */ 23 static const u_int32_t 24 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 25 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 26 27 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ 28 static const double 29 P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ 30 P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ 31 P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ 32 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ 33 P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ 34 35 double 36 cbrt(double x) 37 { 38 int32_t hx; 39 union { 40 double value; 41 uint64_t bits; 42 } u; 43 double r,s,t=0.0,w; 44 u_int32_t sign; 45 u_int32_t high,low; 46 47 EXTRACT_WORDS(hx,low,x); 48 sign=hx&0x80000000; /* sign= sign(x) */ 49 hx ^=sign; 50 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 51 52 /* 53 * Rough cbrt to 5 bits: 54 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 55 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 56 * "%" are integer division and modulus with rounding towards minus 57 * infinity. The RHS is always >= the LHS and has a maximum relative 58 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 59 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 60 * floating point representation, for finite positive normal values, 61 * ordinary integer division of the value in bits magically gives 62 * almost exactly the RHS of the above provided we first subtract the 63 * exponent bias (1023 for doubles) and later add it back. We do the 64 * subtraction virtually to keep e >= 0 so that ordinary integer 65 * division rounds towards minus infinity; this is also efficient. 66 */ 67 if(hx<0x00100000) { /* zero or subnormal? */ 68 if((hx|low)==0) 69 return(x); /* cbrt(0) is itself */ 70 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 71 t*=x; 72 GET_HIGH_WORD(high,t); 73 INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0); 74 } else 75 INSERT_WORDS(t,sign|(hx/3+B1),0); 76 77 /* 78 * New cbrt to 23 bits: 79 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) 80 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) 81 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation 82 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this 83 * gives us bounds for r = t**3/x. 84 * 85 * Try to optimize for parallel evaluation as in k_tanf.c. 86 */ 87 r=(t*t)*(t/x); 88 t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); 89 90 /* 91 * Round t away from zero to 23 bits (sloppily except for ensuring that 92 * the result is larger in magnitude than cbrt(x) but not much more than 93 * 2 23-bit ulps larger). With rounding towards zero, the error bound 94 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps 95 * in the rounded t, the infinite-precision error in the Newton 96 * approximation barely affects third digit in the final error 97 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps 98 * before the final error is larger than 0.667 ulps. 99 */ 100 u.value=t; 101 u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL; 102 t=u.value; 103 104 /* one step Newton iteration to 53 bits with error < 0.667 ulps */ 105 s=t*t; /* t*t is exact */ 106 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 107 w=t+t; /* t+t is exact */ 108 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 109 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ 110 111 return(t); 112 } 113 114 #if (LDBL_MANT_DIG == 53) 115 __weak_reference(cbrt, cbrtl); 116 #endif 117