diff options
Diffstat (limited to 'fpu')
| -rw-r--r-- | fpu/softfloat-parts.c.inc | 206 | ||||
| -rw-r--r-- | fpu/softfloat.c | 207 |
2 files changed, 261 insertions, 152 deletions
diff --git a/fpu/softfloat-parts.c.inc b/fpu/softfloat-parts.c.inc index bf935c4fc2..d69f357352 100644 --- a/fpu/softfloat-parts.c.inc +++ b/fpu/softfloat-parts.c.inc @@ -598,6 +598,212 @@ static FloatPartsN *partsN(div)(FloatPartsN *a, FloatPartsN *b, } /* + * Square Root + * + * The base algorithm is lifted from + * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtf.c + * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrt.c + * https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtl.c + * and is thus MIT licenced. + */ +static void partsN(sqrt)(FloatPartsN *a, float_status *status, + const FloatFmt *fmt) +{ + const uint32_t three32 = 3u << 30; + const uint64_t three64 = 3ull << 62; + uint32_t d32, m32, r32, s32, u32; /* 32-bit computation */ + uint64_t d64, m64, r64, s64, u64; /* 64-bit computation */ + uint64_t dh, dl, rh, rl, sh, sl, uh, ul; /* 128-bit computation */ + uint64_t d0h, d0l, d1h, d1l, d2h, d2l; + uint64_t discard; + bool exp_odd; + size_t index; + + if (unlikely(a->cls != float_class_normal)) { + switch (a->cls) { + case float_class_snan: + case float_class_qnan: + parts_return_nan(a, status); + return; + case float_class_zero: + return; + case float_class_inf: + if (unlikely(a->sign)) { + goto d_nan; + } + return; + default: + g_assert_not_reached(); + } + } + + if (unlikely(a->sign)) { + goto d_nan; + } + + /* + * Argument reduction. + * x = 4^e frac; with integer e, and frac in [1, 4) + * m = frac fixed point at bit 62, since we're in base 4. + * If base-2 exponent is odd, exchange that for multiply by 2, + * which results in no shift. + */ + exp_odd = a->exp & 1; + index = extract64(a->frac_hi, 57, 6) | (!exp_odd << 6); + if (!exp_odd) { + frac_shr(a, 1); + } + + /* + * Approximate r ~= 1/sqrt(m) and s ~= sqrt(m) when m in [1, 4). + * + * Initial estimate: + * 7-bit lookup table (1-bit exponent and 6-bit significand). + * + * The relative error (e = r0*sqrt(m)-1) of a linear estimate + * (r0 = a*m + b) is |e| < 0.085955 ~ 0x1.6p-4 at best; + * a table lookup is faster and needs one less iteration. + * The 7-bit table gives |e| < 0x1.fdp-9. + * + * A Newton-Raphson iteration for r is + * s = m*r + * d = s*r + * u = 3 - d + * r = r*u/2 + * + * Fixed point representations: + * m, s, d, u, three are all 2.30; r is 0.32 + */ + m64 = a->frac_hi; + m32 = m64 >> 32; + + r32 = rsqrt_tab[index] << 16; + /* |r*sqrt(m) - 1| < 0x1.FDp-9 */ + + s32 = ((uint64_t)m32 * r32) >> 32; + d32 = ((uint64_t)s32 * r32) >> 32; + u32 = three32 - d32; + + if (N == 64) { + /* float64 or smaller */ + + r32 = ((uint64_t)r32 * u32) >> 31; + /* |r*sqrt(m) - 1| < 0x1.7Bp-16 */ + + s32 = ((uint64_t)m32 * r32) >> 32; + d32 = ((uint64_t)s32 * r32) >> 32; + u32 = three32 - d32; + + if (fmt->frac_size <= 23) { + /* float32 or smaller */ + + s32 = ((uint64_t)s32 * u32) >> 32; /* 3.29 */ + s32 = (s32 - 1) >> 6; /* 9.23 */ + /* s < sqrt(m) < s + 0x1.08p-23 */ + + /* compute nearest rounded result to 2.23 bits */ + uint32_t d0 = (m32 << 16) - s32 * s32; + uint32_t d1 = s32 - d0; + uint32_t d2 = d1 + s32 + 1; + s32 += d1 >> 31; + a->frac_hi = (uint64_t)s32 << (64 - 25); + + /* increment or decrement for inexact */ + if (d2 != 0) { + a->frac_hi += ((int32_t)(d1 ^ d2) < 0 ? -1 : 1); + } + goto done; + } + + /* float64 */ + + r64 = (uint64_t)r32 * u32 * 2; + /* |r*sqrt(m) - 1| < 0x1.37-p29; convert to 64-bit arithmetic */ + mul64To128(m64, r64, &s64, &discard); + mul64To128(s64, r64, &d64, &discard); + u64 = three64 - d64; + + mul64To128(s64, u64, &s64, &discard); /* 3.61 */ + s64 = (s64 - 2) >> 9; /* 12.52 */ + + /* Compute nearest rounded result */ + uint64_t d0 = (m64 << 42) - s64 * s64; + uint64_t d1 = s64 - d0; + uint64_t d2 = d1 + s64 + 1; + s64 += d1 >> 63; + a->frac_hi = s64 << (64 - 54); + + /* increment or decrement for inexact */ + if (d2 != 0) { + a->frac_hi += ((int64_t)(d1 ^ d2) < 0 ? -1 : 1); + } + goto done; + } + + r64 = (uint64_t)r32 * u32 * 2; + /* |r*sqrt(m) - 1| < 0x1.7Bp-16; convert to 64-bit arithmetic */ + + mul64To128(m64, r64, &s64, &discard); + mul64To128(s64, r64, &d64, &discard); + u64 = three64 - d64; + mul64To128(u64, r64, &r64, &discard); + r64 <<= 1; + /* |r*sqrt(m) - 1| < 0x1.a5p-31 */ + + mul64To128(m64, r64, &s64, &discard); + mul64To128(s64, r64, &d64, &discard); + u64 = three64 - d64; + mul64To128(u64, r64, &rh, &rl); + add128(rh, rl, rh, rl, &rh, &rl); + /* |r*sqrt(m) - 1| < 0x1.c001p-59; change to 128-bit arithmetic */ + + mul128To256(a->frac_hi, a->frac_lo, rh, rl, &sh, &sl, &discard, &discard); + mul128To256(sh, sl, rh, rl, &dh, &dl, &discard, &discard); + sub128(three64, 0, dh, dl, &uh, &ul); + mul128To256(uh, ul, sh, sl, &sh, &sl, &discard, &discard); /* 3.125 */ + /* -0x1p-116 < s - sqrt(m) < 0x3.8001p-125 */ + + sub128(sh, sl, 0, 4, &sh, &sl); + shift128Right(sh, sl, 13, &sh, &sl); /* 16.112 */ + /* s < sqrt(m) < s + 1ulp */ + + /* Compute nearest rounded result */ + mul64To128(sl, sl, &d0h, &d0l); + d0h += 2 * sh * sl; + sub128(a->frac_lo << 34, 0, d0h, d0l, &d0h, &d0l); + sub128(sh, sl, d0h, d0l, &d1h, &d1l); + add128(sh, sl, 0, 1, &d2h, &d2l); + add128(d2h, d2l, d1h, d1l, &d2h, &d2l); + add128(sh, sl, 0, d1h >> 63, &sh, &sl); + shift128Left(sh, sl, 128 - 114, &sh, &sl); + + /* increment or decrement for inexact */ + if (d2h | d2l) { + if ((int64_t)(d1h ^ d2h) < 0) { + sub128(sh, sl, 0, 1, &sh, &sl); + } else { + add128(sh, sl, 0, 1, &sh, &sl); + } + } + a->frac_lo = sl; + a->frac_hi = sh; + + done: + /* Convert back from base 4 to base 2. */ + a->exp >>= 1; + if (!(a->frac_hi & DECOMPOSED_IMPLICIT_BIT)) { + frac_add(a, a, a); + } else { + a->exp += 1; + } + return; + + d_nan: + float_raise(float_flag_invalid, status); + parts_default_nan(a, status); +} + +/* * Rounds the floating-point value `a' to an integer, and returns the * result as a floating-point value. The operation is performed * according to the IEC/IEEE Standard for Binary Floating-Point diff --git a/fpu/softfloat.c b/fpu/softfloat.c index 666b5a25d6..0f2eed8d29 100644 --- a/fpu/softfloat.c +++ b/fpu/softfloat.c @@ -820,6 +820,12 @@ static FloatParts128 *parts128_div(FloatParts128 *a, FloatParts128 *b, #define parts_div(A, B, S) \ PARTS_GENERIC_64_128(div, A)(A, B, S) +static void parts64_sqrt(FloatParts64 *a, float_status *s, const FloatFmt *f); +static void parts128_sqrt(FloatParts128 *a, float_status *s, const FloatFmt *f); + +#define parts_sqrt(A, S, F) \ + PARTS_GENERIC_64_128(sqrt, A)(A, S, F) + static bool parts64_round_to_int_normal(FloatParts64 *a, FloatRoundMode rm, int scale, int frac_size); static bool parts128_round_to_int_normal(FloatParts128 *a, FloatRoundMode r, @@ -1386,6 +1392,30 @@ static void frac128_widen(FloatParts256 *r, FloatParts128 *a) #define frac_widen(A, B) FRAC_GENERIC_64_128(widen, B)(A, B) +/* + * Reciprocal sqrt table. 1 bit of exponent, 6-bits of mantessa. + * From https://git.musl-libc.org/cgit/musl/tree/src/math/sqrt_data.c + * and thus MIT licenced. + */ +static const uint16_t rsqrt_tab[128] = { + 0xb451, 0xb2f0, 0xb196, 0xb044, 0xaef9, 0xadb6, 0xac79, 0xab43, + 0xaa14, 0xa8eb, 0xa7c8, 0xa6aa, 0xa592, 0xa480, 0xa373, 0xa26b, + 0xa168, 0xa06a, 0x9f70, 0x9e7b, 0x9d8a, 0x9c9d, 0x9bb5, 0x9ad1, + 0x99f0, 0x9913, 0x983a, 0x9765, 0x9693, 0x95c4, 0x94f8, 0x9430, + 0x936b, 0x92a9, 0x91ea, 0x912e, 0x9075, 0x8fbe, 0x8f0a, 0x8e59, + 0x8daa, 0x8cfe, 0x8c54, 0x8bac, 0x8b07, 0x8a64, 0x89c4, 0x8925, + 0x8889, 0x87ee, 0x8756, 0x86c0, 0x862b, 0x8599, 0x8508, 0x8479, + 0x83ec, 0x8361, 0x82d8, 0x8250, 0x81c9, 0x8145, 0x80c2, 0x8040, + 0xff02, 0xfd0e, 0xfb25, 0xf947, 0xf773, 0xf5aa, 0xf3ea, 0xf234, + 0xf087, 0xeee3, 0xed47, 0xebb3, 0xea27, 0xe8a3, 0xe727, 0xe5b2, + 0xe443, 0xe2dc, 0xe17a, 0xe020, 0xdecb, 0xdd7d, 0xdc34, 0xdaf1, + 0xd9b3, 0xd87b, 0xd748, 0xd61a, 0xd4f1, 0xd3cd, 0xd2ad, 0xd192, + 0xd07b, 0xcf69, 0xce5b, 0xcd51, 0xcc4a, 0xcb48, 0xca4a, 0xc94f, + 0xc858, 0xc764, 0xc674, 0xc587, 0xc49d, 0xc3b7, 0xc2d4, 0xc1f4, + 0xc116, 0xc03c, 0xbf65, 0xbe90, 0xbdbe, 0xbcef, 0xbc23, 0xbb59, + 0xba91, 0xb9cc, 0xb90a, 0xb84a, 0xb78c, 0xb6d0, 0xb617, 0xb560, +}; + #define partsN(NAME) glue(glue(glue(parts,N),_),NAME) #define FloatPartsN glue(FloatParts,N) #define FloatPartsW glue(FloatParts,W) @@ -3586,103 +3616,35 @@ float128 float128_scalbn(float128 a, int n, float_status *status) /* * Square Root - * - * The old softfloat code did an approximation step before zeroing in - * on the final result. However for simpleness we just compute the - * square root by iterating down from the implicit bit to enough extra - * bits to ensure we get a correctly rounded result. - * - * This does mean however the calculation is slower than before, - * especially for 64 bit floats. */ -static FloatParts64 sqrt_float(FloatParts64 a, float_status *s, const FloatFmt *p) -{ - uint64_t a_frac, r_frac, s_frac; - int bit, last_bit; - - if (is_nan(a.cls)) { - parts_return_nan(&a, s); - return a; - } - if (a.cls == float_class_zero) { - return a; /* sqrt(+-0) = +-0 */ - } - if (a.sign) { - float_raise(float_flag_invalid, s); - parts_default_nan(&a, s); - return a; - } - if (a.cls == float_class_inf) { - return a; /* sqrt(+inf) = +inf */ - } - - assert(a.cls == float_class_normal); - - /* We need two overflow bits at the top. Adding room for that is a - * right shift. If the exponent is odd, we can discard the low bit - * by multiplying the fraction by 2; that's a left shift. Combine - * those and we shift right by 1 if the exponent is odd, otherwise 2. - */ - a_frac = a.frac >> (2 - (a.exp & 1)); - a.exp >>= 1; - - /* Bit-by-bit computation of sqrt. */ - r_frac = 0; - s_frac = 0; - - /* Iterate from implicit bit down to the 3 extra bits to compute a - * properly rounded result. Remember we've inserted two more bits - * at the top, so these positions are two less. - */ - bit = DECOMPOSED_BINARY_POINT - 2; - last_bit = MAX(p->frac_shift - 4, 0); - do { - uint64_t q = 1ULL << bit; - uint64_t t_frac = s_frac + q; - if (t_frac <= a_frac) { - s_frac = t_frac + q; - a_frac -= t_frac; - r_frac += q; - } - a_frac <<= 1; - } while (--bit >= last_bit); - - /* Undo the right shift done above. If there is any remaining - * fraction, the result is inexact. Set the sticky bit. - */ - a.frac = (r_frac << 2) + (a_frac != 0); - - return a; -} - float16 QEMU_FLATTEN float16_sqrt(float16 a, float_status *status) { - FloatParts64 pa, pr; + FloatParts64 p; - float16_unpack_canonical(&pa, a, status); - pr = sqrt_float(pa, status, &float16_params); - return float16_round_pack_canonical(&pr, status); + float16_unpack_canonical(&p, a, status); + parts_sqrt(&p, status, &float16_params); + return float16_round_pack_canonical(&p, status); } static float32 QEMU_SOFTFLOAT_ATTR soft_f32_sqrt(float32 a, float_status *status) { - FloatParts64 pa, pr; + FloatParts64 p; - float32_unpack_canonical(&pa, a, status); - pr = sqrt_float(pa, status, &float32_params); - return float32_round_pack_canonical(&pr, status); + float32_unpack_canonical(&p, a, status); + parts_sqrt(&p, status, &float32_params); + return float32_round_pack_canonical(&p, status); } static float64 QEMU_SOFTFLOAT_ATTR soft_f64_sqrt(float64 a, float_status *status) { - FloatParts64 pa, pr; + FloatParts64 p; - float64_unpack_canonical(&pa, a, status); - pr = sqrt_float(pa, status, &float64_params); - return float64_round_pack_canonical(&pr, status); + float64_unpack_canonical(&p, a, status); + parts_sqrt(&p, status, &float64_params); + return float64_round_pack_canonical(&p, status); } float32 QEMU_FLATTEN float32_sqrt(float32 xa, float_status *s) @@ -3741,11 +3703,20 @@ float64 QEMU_FLATTEN float64_sqrt(float64 xa, float_status *s) bfloat16 QEMU_FLATTEN bfloat16_sqrt(bfloat16 a, float_status *status) { - FloatParts64 pa, pr; + FloatParts64 p; - bfloat16_unpack_canonical(&pa, a, status); - pr = sqrt_float(pa, status, &bfloat16_params); - return bfloat16_round_pack_canonical(&pr, status); + bfloat16_unpack_canonical(&p, a, status); + parts_sqrt(&p, status, &bfloat16_params); + return bfloat16_round_pack_canonical(&p, status); +} + +float128 QEMU_FLATTEN float128_sqrt(float128 a, float_status *status) +{ + FloatParts128 p; + + float128_unpack_canonical(&p, a, status); + parts_sqrt(&p, status, &float128_params); + return float128_round_pack_canonical(&p, status); } /*---------------------------------------------------------------------------- @@ -6473,74 +6444,6 @@ float128 float128_rem(float128 a, float128 b, float_status *status) status); } -/*---------------------------------------------------------------------------- -| Returns the square root of the quadruple-precision floating-point value `a'. -| The operation is performed according to the IEC/IEEE Standard for Binary -| Floating-Point Arithmetic. -*----------------------------------------------------------------------------*/ - -float128 float128_sqrt(float128 a, float_status *status) -{ - bool aSign; - int32_t aExp, zExp; - uint64_t aSig0, aSig1, zSig0, zSig1, zSig2, doubleZSig0; - uint64_t rem0, rem1, rem2, rem3, term0, term1, term2, term3; - - aSig1 = extractFloat128Frac1( a ); - aSig0 = extractFloat128Frac0( a ); - aExp = extractFloat128Exp( a ); - aSign = extractFloat128Sign( a ); - if ( aExp == 0x7FFF ) { - if (aSig0 | aSig1) { - return propagateFloat128NaN(a, a, status); - } - if ( ! aSign ) return a; - goto invalid; - } - if ( aSign ) { - if ( ( aExp | aSig0 | aSig1 ) == 0 ) return a; - invalid: - float_raise(float_flag_invalid, status); - return float128_default_nan(status); - } - if ( aExp == 0 ) { - if ( ( aSig0 | aSig1 ) == 0 ) return packFloat128( 0, 0, 0, 0 ); - normalizeFloat128Subnormal( aSig0, aSig1, &aExp, &aSig0, &aSig1 ); - } - zExp = ( ( aExp - 0x3FFF )>>1 ) + 0x3FFE; - aSig0 |= UINT64_C(0x0001000000000000); - zSig0 = estimateSqrt32( aExp, aSig0>>17 ); - shortShift128Left( aSig0, aSig1, 13 - ( aExp & 1 ), &aSig0, &aSig1 ); - zSig0 = estimateDiv128To64( aSig0, aSig1, zSig0<<32 ) + ( zSig0<<30 ); - doubleZSig0 = zSig0<<1; - mul64To128( zSig0, zSig0, &term0, &term1 ); - sub128( aSig0, aSig1, term0, term1, &rem0, &rem1 ); - while ( (int64_t) rem0 < 0 ) { - --zSig0; - doubleZSig0 -= 2; - add128( rem0, rem1, zSig0>>63, doubleZSig0 | 1, &rem0, &rem1 ); - } - zSig1 = estimateDiv128To64( rem1, 0, doubleZSig0 ); - if ( ( zSig1 & 0x1FFF ) <= 5 ) { - if ( zSig1 == 0 ) zSig1 = 1; - mul64To128( doubleZSig0, zSig1, &term1, &term2 ); - sub128( rem1, 0, term1, term2, &rem1, &rem2 ); - mul64To128( zSig1, zSig1, &term2, &term3 ); - sub192( rem1, rem2, 0, 0, term2, term3, &rem1, &rem2, &rem3 ); - while ( (int64_t) rem1 < 0 ) { - --zSig1; - shortShift128Left( 0, zSig1, 1, &term2, &term3 ); - term3 |= 1; - term2 |= doubleZSig0; - add192( rem1, rem2, rem3, 0, term2, term3, &rem1, &rem2, &rem3 ); - } - zSig1 |= ( ( rem1 | rem2 | rem3 ) != 0 ); - } - shift128ExtraRightJamming( zSig0, zSig1, 0, 14, &zSig0, &zSig1, &zSig2 ); - return roundAndPackFloat128(0, zExp, zSig0, zSig1, zSig2, status); - -} - static inline FloatRelation floatx80_compare_internal(floatx80 a, floatx80 b, bool is_quiet, float_status *status) |