/*============================================================================ This source file is an extension to the SoftFloat IEC/IEEE Floating-point Arithmetic Package, Release 2b, written for Bochs (x86 achitecture simulator) floating point emulation. THIS SOFTWARE IS DISTRIBUTED AS IS, FOR FREE. Although reasonable effort has been made to avoid it, THIS SOFTWARE MAY CONTAIN FAULTS THAT WILL AT TIMES RESULT IN INCORRECT BEHAVIOR. USE OF THIS SOFTWARE IS RESTRICTED TO PERSONS AND ORGANIZATIONS WHO CAN AND WILL TAKE FULL RESPONSIBILITY FOR ALL LOSSES, COSTS, OR OTHER PROBLEMS THEY INCUR DUE TO THE SOFTWARE, AND WHO FURTHERMORE EFFECTIVELY INDEMNIFY JOHN HAUSER AND THE INTERNATIONAL COMPUTER SCIENCE INSTITUTE (possibly via similar legal warning) AGAINST ALL LOSSES, COSTS, OR OTHER PROBLEMS INCURRED BY THEIR CUSTOMERS AND CLIENTS DUE TO THE SOFTWARE. Derivative works are acceptable, even for commercial purposes, so long as (1) the source code for the derivative work includes prominent notice that the work is derivative, and (2) the source code includes prominent notice with these four paragraphs for those parts of this code that are retained. =============================================================================*/ /*============================================================================ * Written for Bochs (x86 achitecture simulator) by * Stanislav Shwartsman [sshwarts at sourceforge net] * ==========================================================================*/ #define FLOAT128 #define USE_estimateDiv128To64 #define FLOAT128 #include "../build/MAME/platform.h" #include "../source/include/internals.h" #include "../source/include/softfloat.h" #include "../source/8086/specialize.h" #include "fpu_constant.h" #include "softfloat-extra.h" #include "softfloat-helpers.h" #include "softfloat-specialize.h" static const extFloat80_t floatx80_one = packFloatx80(0, 0x3fff, uint64_t(0x8000000000000000)); /* reduce trigonometric function argument using 128-bit precision M_PI approximation */ static uint64_t argument_reduction_kernel(uint64_t aSig0, int Exp, uint64_t *zSig0, uint64_t *zSig1) { uint64_t term0, term1, term2; uint64_t aSig1 = 0; shortShift128Left(aSig1, aSig0, Exp, &aSig1, &aSig0); uint64_t q = estimateDiv128To64(aSig1, aSig0, FLOAT_PI_HI); mul128By64To192(FLOAT_PI_HI, FLOAT_PI_LO, q, &term0, &term1, &term2); sub128(aSig1, aSig0, term0, term1, zSig1, zSig0); while ((int64_t)(*zSig1) < 0) { --q; add192(*zSig1, *zSig0, term2, 0, FLOAT_PI_HI, FLOAT_PI_LO, zSig1, zSig0, &term2); } *zSig1 = term2; return q; } static int reduce_trig_arg(int expDiff, int &zSign, uint64_t &aSig0, uint64_t &aSig1) { uint64_t term0, term1, q = 0; if (expDiff < 0) { shortShift128Right(aSig0, 0, 1, &aSig0, &aSig1); expDiff = 0; } if (expDiff > 0) { q = argument_reduction_kernel(aSig0, expDiff, &aSig0, &aSig1); } else { if (FLOAT_PI_HI <= aSig0) { aSig0 -= FLOAT_PI_HI; q = 1; } } shortShift128Right(FLOAT_PI_HI, FLOAT_PI_LO, 1, &term0, &term1); if (! softfloat_lt128(aSig0, aSig1, term0, term1)) { int lt = softfloat_lt128(term0, term1, aSig0, aSig1); int eq = softfloat_eq128(aSig0, aSig1, term0, term1); if ((eq && (q & 1)) || lt) { zSign = !zSign; ++q; } if (lt) sub128(FLOAT_PI_HI, FLOAT_PI_LO, aSig0, aSig1, &aSig0, &aSig1); } return (int)(q & 3); } #define SIN_ARR_SIZE 11 #define COS_ARR_SIZE 11 static float128_t sin_arr[SIN_ARR_SIZE] = { PACK_FLOAT_128(0x3fff000000000000, 0x0000000000000000), /* 1 */ PACK_FLOAT_128(0xbffc555555555555, 0x5555555555555555), /* 3 */ PACK_FLOAT_128(0x3ff8111111111111, 0x1111111111111111), /* 5 */ PACK_FLOAT_128(0xbff2a01a01a01a01, 0xa01a01a01a01a01a), /* 7 */ PACK_FLOAT_128(0x3fec71de3a556c73, 0x38faac1c88e50017), /* 9 */ PACK_FLOAT_128(0xbfe5ae64567f544e, 0x38fe747e4b837dc7), /* 11 */ PACK_FLOAT_128(0x3fde6124613a86d0, 0x97ca38331d23af68), /* 13 */ PACK_FLOAT_128(0xbfd6ae7f3e733b81, 0xf11d8656b0ee8cb0), /* 15 */ PACK_FLOAT_128(0x3fce952c77030ad4, 0xa6b2605197771b00), /* 17 */ PACK_FLOAT_128(0xbfc62f49b4681415, 0x724ca1ec3b7b9675), /* 19 */ PACK_FLOAT_128(0x3fbd71b8ef6dcf57, 0x18bef146fcee6e45) /* 21 */ }; static float128_t cos_arr[COS_ARR_SIZE] = { PACK_FLOAT_128(0x3fff000000000000, 0x0000000000000000), /* 0 */ PACK_FLOAT_128(0xbffe000000000000, 0x0000000000000000), /* 2 */ PACK_FLOAT_128(0x3ffa555555555555, 0x5555555555555555), /* 4 */ PACK_FLOAT_128(0xbff56c16c16c16c1, 0x6c16c16c16c16c17), /* 6 */ PACK_FLOAT_128(0x3fefa01a01a01a01, 0xa01a01a01a01a01a), /* 8 */ PACK_FLOAT_128(0xbfe927e4fb7789f5, 0xc72ef016d3ea6679), /* 10 */ PACK_FLOAT_128(0x3fe21eed8eff8d89, 0x7b544da987acfe85), /* 12 */ PACK_FLOAT_128(0xbfda93974a8c07c9, 0xd20badf145dfa3e5), /* 14 */ PACK_FLOAT_128(0x3fd2ae7f3e733b81, 0xf11d8656b0ee8cb0), /* 16 */ PACK_FLOAT_128(0xbfca6827863b97d9, 0x77bb004886a2c2ab), /* 18 */ PACK_FLOAT_128(0x3fc1e542ba402022, 0x507a9cad2bf8f0bb) /* 20 */ }; extern float128_t OddPoly (float128_t x, const float128_t *arr, int n); /* 0 <= x <= pi/4 */ inline float128_t poly_sin(float128_t x) { // 3 5 7 9 11 13 15 // x x x x x x x // sin (x) ~ x - --- + --- - --- + --- - ---- + ---- - ---- = // 3! 5! 7! 9! 11! 13! 15! // // 2 4 6 8 10 12 14 // x x x x x x x // = x * [ 1 - --- + --- - --- + --- - ---- + ---- - ---- ] = // 3! 5! 7! 9! 11! 13! 15! // // 3 3 // -- 4k -- 4k+2 // p(x) = > C * x > 0 q(x) = > C * x < 0 // -- 2k -- 2k+1 // k=0 k=0 // // 2 // sin(x) ~ x * [ p(x) + x * q(x) ] // return OddPoly(x, (const float128_t*) sin_arr, SIN_ARR_SIZE); } extern float128_t EvenPoly(float128_t x, const float128_t *arr, int n); /* 0 <= x <= pi/4 */ inline float128_t poly_cos(float128_t x) { // 2 4 6 8 10 12 14 // x x x x x x x // cos (x) ~ 1 - --- + --- - --- + --- - ---- + ---- - ---- // 2! 4! 6! 8! 10! 12! 14! // // 3 3 // -- 4k -- 4k+2 // p(x) = > C * x > 0 q(x) = > C * x < 0 // -- 2k -- 2k+1 // k=0 k=0 // // 2 // cos(x) ~ [ p(x) + x * q(x) ] // return EvenPoly(x, (const float128_t*) cos_arr, COS_ARR_SIZE); } inline void sincos_invalid(extFloat80_t *sin_a, extFloat80_t *cos_a, extFloat80_t a) { if (sin_a) *sin_a = a; if (cos_a) *cos_a = a; } inline void sincos_tiny_argument(extFloat80_t *sin_a, extFloat80_t *cos_a, extFloat80_t a) { if (sin_a) *sin_a = a; if (cos_a) *cos_a = floatx80_one; } static extFloat80_t sincos_approximation(int neg, float128_t r, uint64_t quotient) { if (quotient & 0x1) { r = poly_cos(r); neg = 0; } else { r = poly_sin(r); } extFloat80_t result = f128_to_extF80(r); if (quotient & 0x2) neg = ! neg; if (neg) floatx80_chs(result); return result; } // ================================================= // FSINCOS Compute sin(x) and cos(x) // ================================================= // // Uses the following identities: // ---------------------------------------------------------- // // sin(-x) = -sin(x) // cos(-x) = cos(x) // // sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y) // cos(x+y) = sin(x)*sin(y)+cos(x)*cos(y) // // sin(x+ pi/2) = cos(x) // sin(x+ pi) = -sin(x) // sin(x+3pi/2) = -cos(x) // sin(x+2pi) = sin(x) // int extFloat80_sincos(extFloat80_t a, extFloat80_t *sin_a, extFloat80_t *cos_a) { uint64_t aSig0, aSig1 = 0; int32_t aExp, zExp, expDiff; int aSign, zSign; int q = 0; // handle unsupported extended double-precision floating encodings if (extF80_isUnsupported(a)) { goto invalid; } aSig0 = extF80_fraction(a); aExp = extF80_exp(a); aSign = extF80_sign(a); /* invalid argument */ if (aExp == 0x7FFF) { if (aSig0 << 1) { const uint128 nan = softfloat_propagateNaNExtF80UI(a.signExp, aSig0, 0, 0); extFloat80_t rv; rv.signExp = nan.v64; rv.signif = nan.v0; sincos_invalid(sin_a, cos_a, rv); return 0; } invalid: softfloat_exceptionFlags |= softfloat_flag_invalid; sincos_invalid(sin_a, cos_a, floatx80_default_nan); return 0; } if (! aExp) { if (! aSig0) { sincos_tiny_argument(sin_a, cos_a, a); return 0; } softfloat_exceptionFlags |= softfloat_flag_invalid; // actually denormal /* handle pseudo denormals */ if (! (aSig0 & uint64_t(0x8000000000000000))) { softfloat_exceptionFlags |= softfloat_flag_inexact; if (sin_a) softfloat_exceptionFlags |= softfloat_flag_underflow; sincos_tiny_argument(sin_a, cos_a, a); return 0; } struct exp32_sig64 normExpSig = softfloat_normSubnormalExtF80Sig(aSig0); aExp = normExpSig.exp + 1; aSig0 = normExpSig.sig; } zSign = aSign; zExp = FLOATX80_EXP_BIAS; expDiff = aExp - zExp; /* argument is out-of-range */ if (expDiff >= 63) return -1; softfloat_exceptionFlags |= softfloat_flag_inexact; if (expDiff < -1) { // doesn't require reduction if (expDiff <= -68) { a = packFloatx80(aSign, aExp, aSig0); sincos_tiny_argument(sin_a, cos_a, a); return 0; } zExp = aExp; } else { q = reduce_trig_arg(expDiff, zSign, aSig0, aSig1); } /* **************************** */ /* argument reduction completed */ /* **************************** */ /* using float128 for approximation */ float128_t r = softfloat_normRoundPackToF128(0, zExp-0x10, aSig0, aSig1); if (aSign) q = -q; if (sin_a) *sin_a = sincos_approximation(zSign, r, q); if (cos_a) *cos_a = sincos_approximation(zSign, r, q+1); return 0; } int extFloat80_sin(extFloat80_t &a) { return extFloat80_sincos(a, &a, 0); } int extFloat80_cos(extFloat80_t &a) { return extFloat80_sincos(a, 0, &a); } // ================================================= // FPTAN Compute tan(x) // ================================================= // // Uses the following identities: // // 1. ---------------------------------------------------------- // // sin(-x) = -sin(x) // cos(-x) = cos(x) // // sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y) // cos(x+y) = sin(x)*sin(y)+cos(x)*cos(y) // // sin(x+ pi/2) = cos(x) // sin(x+ pi) = -sin(x) // sin(x+3pi/2) = -cos(x) // sin(x+2pi) = sin(x) // // 2. ---------------------------------------------------------- // // sin(x) // tan(x) = ------ // cos(x) // int extFloat80_tan(extFloat80_t &a) { uint64_t aSig0, aSig1 = 0; int32_t aExp, zExp, expDiff; int aSign, zSign; int q = 0; // handle unsupported extended double-precision floating encodings if (extF80_isUnsupported(a)) { goto invalid; } aSig0 = extF80_fraction(a); aExp = extF80_exp(a); aSign = extF80_sign(a); /* invalid argument */ if (aExp == 0x7FFF) { if (aSig0 << 1) { const uint128 nan = softfloat_propagateNaNExtF80UI(a.signExp, aSig0, 0, 0); a.signExp = nan.v64; a.signif = nan.v0; return 0; } invalid: softfloat_exceptionFlags |= softfloat_flag_invalid; a = floatx80_default_nan; return 0; } if (! aExp) { if (! aSig0) return 0; softfloat_exceptionFlags |= softfloat_flag_invalid; // actually denormal /* handle pseudo denormals */ if (! (aSig0 & uint64_t(0x8000000000000000))) { softfloat_exceptionFlags |= softfloat_flag_inexact | softfloat_flag_underflow; return 0; } struct exp32_sig64 normExpSig = softfloat_normSubnormalExtF80Sig(aSig0); aExp = normExpSig.exp + 1; aSig0 = normExpSig.sig; } zSign = aSign; zExp = FLOATX80_EXP_BIAS; expDiff = aExp - zExp; /* argument is out-of-range */ if (expDiff >= 63) return -1; softfloat_exceptionFlags |= softfloat_flag_inexact; if (expDiff < -1) { // doesn't require reduction if (expDiff <= -68) { a = packFloatx80(aSign, aExp, aSig0); return 0; } zExp = aExp; } else { q = reduce_trig_arg(expDiff, zSign, aSig0, aSig1); } /* **************************** */ /* argument reduction completed */ /* **************************** */ /* using float128 for approximation */ float128_t r = softfloat_normRoundPackToF128(0, zExp-0x10, aSig0, aSig1); float128_t sin_r = poly_sin(r); float128_t cos_r = poly_cos(r); if (q & 0x1) { r = f128_div(cos_r, sin_r); zSign = ! zSign; } else { r = f128_div(sin_r, cos_r); } a = f128_to_extF80(r); if (zSign) floatx80_chs(a); return 0; } extFloat80_t &floatx80_chs(extFloat80_t ®) { reg.signExp ^= 0x8000; return reg; }