1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 * 11 * Optimized by Bruce D. Evans. 12 */ 13 14 #include <sys/cdefs.h> 15 #include <float.h> 16 #include "math.h" 17 #include "math_private.h" 18 19 /* cbrt(x) 20 * Return cube root of x 21 */ 22 static const u_int32_t 23 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 24 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 25 26 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ 27 static const double 28 P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ 29 P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ 30 P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ 31 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ 32 P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ 33 34 double 35 cbrt(double x) 36 { 37 int32_t hx; 38 union { 39 double value; 40 uint64_t bits; 41 } u; 42 double r,s,t=0.0,w; 43 u_int32_t sign; 44 u_int32_t high,low; 45 46 EXTRACT_WORDS(hx,low,x); 47 sign=hx&0x80000000; /* sign= sign(x) */ 48 hx ^=sign; 49 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 50 51 /* 52 * Rough cbrt to 5 bits: 53 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 54 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 55 * "%" are integer division and modulus with rounding towards minus 56 * infinity. The RHS is always >= the LHS and has a maximum relative 57 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 58 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 59 * floating point representation, for finite positive normal values, 60 * ordinary integer division of the value in bits magically gives 61 * almost exactly the RHS of the above provided we first subtract the 62 * exponent bias (1023 for doubles) and later add it back. We do the 63 * subtraction virtually to keep e >= 0 so that ordinary integer 64 * division rounds towards minus infinity; this is also efficient. 65 */ 66 if(hx<0x00100000) { /* zero or subnormal? */ 67 if((hx|low)==0) 68 return(x); /* cbrt(0) is itself */ 69 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 70 t*=x; 71 GET_HIGH_WORD(high,t); 72 INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0); 73 } else 74 INSERT_WORDS(t,sign|(hx/3+B1),0); 75 76 /* 77 * New cbrt to 23 bits: 78 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) 79 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) 80 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation 81 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this 82 * gives us bounds for r = t**3/x. 83 * 84 * Try to optimize for parallel evaluation as in k_tanf.c. 85 */ 86 r=(t*t)*(t/x); 87 t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); 88 89 /* 90 * Round t away from zero to 23 bits (sloppily except for ensuring that 91 * the result is larger in magnitude than cbrt(x) but not much more than 92 * 2 23-bit ulps larger). With rounding towards zero, the error bound 93 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps 94 * in the rounded t, the infinite-precision error in the Newton 95 * approximation barely affects third digit in the final error 96 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps 97 * before the final error is larger than 0.667 ulps. 98 */ 99 u.value=t; 100 u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL; 101 t=u.value; 102 103 /* one step Newton iteration to 53 bits with error < 0.667 ulps */ 104 s=t*t; /* t*t is exact */ 105 r=x/s; /* error <= 0.5 ulps; |r| < |t| */ 106 w=t+t; /* t+t is exact */ 107 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ 108 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ 109 110 return(t); 111 } 112 113 #if (LDBL_MANT_DIG == 53) 114 __weak_reference(cbrt, cbrtl); 115 #endif 116