/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * * Optimized by Bruce D. Evans. */ #include "libc/math.h" asm(".ident\t\"\\n\\n\ fdlibm\\n\ Copyright 1993 Sun Microsystems, Inc.\\n\ Developed at SunPro, a Sun Microsystems, Inc. business.\""); static const uint32_t B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ /** * Returns cube root of 𝑥. */ double __cbrt(double x) { union { double f; uint64_t i; } u = {x}; double_t r, s, t, w; uint32_t hx = u.i >> 32 & 0x7fffffff; if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */ return x + x; /* * Rough cbrt to 5 bits: * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) * where e is integral and >= 0, m is real and in [0, 1), and "/" and * "%" are integer division and modulus with rounding towards minus * infinity. The RHS is always >= the LHS and has a maximum relative * error of about 1 in 16. Adding a bias of -0.03306235651 to the * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE * floating point representation, for finite positive normal values, * ordinary integer divison of the value in bits magically gives * almost exactly the RHS of the above provided we first subtract the * exponent bias (1023 for doubles) and later add it back. We do the * subtraction virtually to keep e >= 0 so that ordinary integer * division rounds towards minus infinity; this is also efficient. */ if (hx < 0x00100000) { /* zero or subnormal? */ u.f = x * 0x1p54; hx = u.i >> 32 & 0x7fffffff; if (hx == 0) return x; /* cbrt(0) is itself */ hx = hx / 3 + B2; } else hx = hx / 3 + B1; u.i &= 1ULL << 63; u.i |= (uint64_t)hx << 32; t = u.f; /* * New cbrt to 23 bits: * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this * gives us bounds for r = t**3/x. * * Try to optimize for parallel evaluation as in __tanf.c. */ r = (t * t) * (t / x); t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4)); /* * Round t away from zero to 23 bits (sloppily except for ensuring that * the result is larger in magnitude than cbrt(x) but not much more than * 2 23-bit ulps larger). With rounding towards zero, the error bound * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps * in the rounded t, the infinite-precision error in the Newton * approximation barely affects third digit in the final error * 0.667; the error in the rounded t can be up to about 3 23-bit ulps * before the final error is larger than 0.667 ulps. */ u.f = t; u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL; t = u.f; /* one step Newton iteration to 53 bits with error < 0.667 ulps */ s = t * t; /* t*t is exact */ r = x / s; /* error <= 0.5 ulps; |r| < |t| */ w = t + t; /* t+t is exact */ r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */ t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */ return t; }