/* cbrtf.c * * Cube root * * * * SYNOPSIS: * * float x, y, cbrtf(); * * y = cbrtf( x ); * * * * DESCRIPTION: * * Returns the cube root of the argument, which may be negative. * * Range reduction involves determining the power of 2 of * the argument. A polynomial of degree 2 applied to the * mantissa, and multiplication by the cube root of 1, 2, or 4 * approximates the root to within about 0.1%. Then Newton's * iteration is used to converge to an accurate result. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,1e38 100000 7.6e-8 2.7e-8 * */ /* cbrt.c */ /* Cephes Math Library Release 2.2: June, 1992 Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ #include <math.h> static float CBRT2 = 1.25992104989487316477; static float CBRT4 = 1.58740105196819947475; float frexpf(float, int *), ldexpf(float, int); float cbrtf( float xx ) { int e, rem, sign; float x, z; x = xx; if( x == 0 ) return( 0.0 ); if( x > 0 ) sign = 1; else { sign = -1; x = -x; } z = x; /* extract power of 2, leaving * mantissa between 0.5 and 1 */ x = frexpf( x, &e ); /* Approximate cube root of number between .5 and 1, * peak relative error = 9.2e-6 */ x = (((-0.13466110473359520655053 * x + 0.54664601366395524503440 ) * x - 0.95438224771509446525043 ) * x + 1.1399983354717293273738 ) * x + 0.40238979564544752126924; /* exponent divided by 3 */ if( e >= 0 ) { rem = e; e /= 3; rem -= 3*e; if( rem == 1 ) x *= CBRT2; else if( rem == 2 ) x *= CBRT4; } /* argument less than 1 */ else { e = -e; rem = e; e /= 3; rem -= 3*e; if( rem == 1 ) x /= CBRT2; else if( rem == 2 ) x /= CBRT4; e = -e; } /* multiply by power of 2 */ x = ldexpf( x, e ); /* Newton iteration */ x -= ( x - (z/(x*x)) ) * 0.333333333333; if( sign < 0 ) x = -x; return(x); }