/* Double-precision floating point square root. Copyright (C) 1997, 2002, 2003, 2004 Free Software Foundation, Inc. This file is part of the GNU C Library. The GNU C Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The GNU C Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU C Library; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA. */ #include #include #include #include #include #include static const double almost_half = 0.5000000000000001; /* 0.5 + 2^-53 */ static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 }; static const ieee_float_shape_type a_inf = {.word = 0x7f800000 }; static const float two108 = 3.245185536584267269e+32; static const float twom54 = 5.551115123125782702e-17; extern const float __t_sqrt[1024]; /* The method is based on a description in Computation of elementary functions on the IBM RISC System/6000 processor, P. W. Markstein, IBM J. Res. Develop, 34(1) 1990. Basically, it consists of two interleaved Newton-Rhapson approximations, one to find the actual square root, and one to find its reciprocal without the expense of a division operation. The tricky bit here is the use of the POWER/PowerPC multiply-add operation to get the required accuracy with high speed. The argument reduction works by a combination of table lookup to obtain the initial guesses, and some careful modification of the generated guesses (which mostly runs on the integer unit, while the Newton-Rhapson is running on the FPU). */ #ifdef __STDC__ double __slow_ieee754_sqrt (double x) #else double __slow_ieee754_sqrt (x) double x; #endif { const float inf = a_inf.value; if (x > 0) { /* schedule the EXTRACT_WORDS to get separation between the store and the load. */ ieee_double_shape_type ew_u; ieee_double_shape_type iw_u; ew_u.value = (x); if (x != inf) { /* Variables named starting with 's' exist in the argument-reduced space, so that 2 > sx >= 0.5, 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... . Variables named ending with 'i' are integer versions of floating-point values. */ double sx; /* The value of which we're trying to find the square root. */ double sg, g; /* Guess of the square root of x. */ double sd, d; /* Difference between the square of the guess and x. */ double sy; /* Estimate of 1/2g (overestimated by 1ulp). */ double sy2; /* 2*sy */ double e; /* Difference between y*g and 1/2 (se = e * fsy). */ double shx; /* == sx * fsg */ double fsg; /* sg*fsg == g. */ fenv_t fe; /* Saved floating-point environment (stores rounding mode and whether the inexact exception is enabled). */ uint32_t xi0, xi1, sxi, fsgi; const float *t_sqrt; fe = fegetenv_register (); /* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */ xi0 = ew_u.parts.msw; xi1 = ew_u.parts.lsw; relax_fenv_state (); sxi = (xi0 & 0x3fffffff) | 0x3fe00000; /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation between the store and the load. */ iw_u.parts.msw = sxi; iw_u.parts.lsw = xi1; t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe); sg = t_sqrt[0]; sy = t_sqrt[1]; /* complete the INSERT_WORDS (sx, sxi, xi1) operation. */ sx = iw_u.value; /* Here we have three Newton-Rhapson iterations each of a division and a square root and the remainder of the argument reduction, all interleaved. */ sd = -(sg * sg - sx); fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000; sy2 = sy + sy; sg = sy * sd + sg; /* 16-bit approximation to sqrt(sx). */ /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation between the store and the load. */ INSERT_WORDS (fsg, fsgi, 0); iw_u.parts.msw = fsgi; iw_u.parts.lsw = (0); e = -(sy * sg - almost_half); sd = -(sg * sg - sx); if ((xi0 & 0x7ff00000) == 0) goto denorm; sy = sy + e * sy2; sg = sg + sy * sd; /* 32-bit approximation to sqrt(sx). */ sy2 = sy + sy; /* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */ fsg = iw_u.value; e = -(sy * sg - almost_half); sd = -(sg * sg - sx); sy = sy + e * sy2; shx = sx * fsg; sg = sg + sy * sd; /* 64-bit approximation to sqrt(sx), but perhaps rounded incorrectly. */ sy2 = sy + sy; g = sg * fsg; e = -(sy * sg - almost_half); d = -(g * sg - shx); sy = sy + e * sy2; fesetenv_register (fe); return g + sy * d; denorm: /* For denormalised numbers, we normalise, calculate the square root, and return an adjusted result. */ fesetenv_register (fe); return __slow_ieee754_sqrt (x * two108) * twom54; } } else if (x < 0) { /* For some reason, some PowerPC32 processors don't implement FE_INVALID_SQRT. */ #ifdef FE_INVALID_SQRT feraiseexcept (FE_INVALID_SQRT); if (!fetestexcept (FE_INVALID)) #endif feraiseexcept (FE_INVALID); x = a_nan.value; } return f_wash (x); } #ifdef __STDC__ double __ieee754_sqrt (double x) #else double __ieee754_sqrt (x) double x; #endif { double z; /* If the CPU is 64-bit we can use the optional FP instructions. */ if (__CPU_HAS_FSQRT) { /* Volatile is required to prevent the compiler from moving the fsqrt instruction above the branch. */ __asm __volatile (" fsqrt %0,%1\n" :"=f" (z):"f" (x)); } else z = __slow_ieee754_sqrt (x); return z; }