softfloat: Move sqrt_float to softfloat-parts.c.inc
Rename to parts$N_sqrt. Reimplement float128_sqrt with FloatParts128. Reimplement with the inverse sqrt newton-raphson algorithm from musl. This is significantly faster than even the berkeley sqrt n-r algorithm, because it does not use division instructions, only multiplication. Ordinarily, changing algorithms at the same time as migrating code is a bad idea, but this is the only way I found that didn't break one of the routines at the same time. Tested-by: Alex Bennée <alex.bennee@linaro.org> Reviewed-by: Alex Bennée <alex.bennee@linaro.org> Signed-off-by: Richard Henderson <richard.henderson@linaro.org>
This commit is contained in:
Richard Henderson
parent
39626b0ce8
commit
9261b245f0
@ -597,6 +597,212 @@ static FloatPartsN *partsN(div)(FloatPartsN *a, FloatPartsN *b,
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return a;
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}
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/*
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* Square Root
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*
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* The base algorithm is lifted from
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* https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtf.c
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* https://git.musl-libc.org/cgit/musl/tree/src/math/sqrt.c
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* https://git.musl-libc.org/cgit/musl/tree/src/math/sqrtl.c
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* and is thus MIT licenced.
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*/
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static void partsN(sqrt)(FloatPartsN *a, float_status *status,
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const FloatFmt *fmt)
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{
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const uint32_t three32 = 3u << 30;
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const uint64_t three64 = 3ull << 62;
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uint32_t d32, m32, r32, s32, u32; /* 32-bit computation */
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uint64_t d64, m64, r64, s64, u64; /* 64-bit computation */
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uint64_t dh, dl, rh, rl, sh, sl, uh, ul; /* 128-bit computation */
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uint64_t d0h, d0l, d1h, d1l, d2h, d2l;
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uint64_t discard;
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bool exp_odd;
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size_t index;
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if (unlikely(a->cls != float_class_normal)) {
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switch (a->cls) {
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case float_class_snan:
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case float_class_qnan:
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parts_return_nan(a, status);
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return;
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case float_class_zero:
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return;
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case float_class_inf:
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if (unlikely(a->sign)) {
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goto d_nan;
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}
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return;
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default:
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g_assert_not_reached();
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}
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}
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if (unlikely(a->sign)) {
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goto d_nan;
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}
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/*
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* Argument reduction.
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* x = 4^e frac; with integer e, and frac in [1, 4)
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* m = frac fixed point at bit 62, since we're in base 4.
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* If base-2 exponent is odd, exchange that for multiply by 2,
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* which results in no shift.
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*/
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exp_odd = a->exp & 1;
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index = extract64(a->frac_hi, 57, 6) | (!exp_odd << 6);
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if (!exp_odd) {
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frac_shr(a, 1);
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}
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/*
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* Approximate r ~= 1/sqrt(m) and s ~= sqrt(m) when m in [1, 4).
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*
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* Initial estimate:
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* 7-bit lookup table (1-bit exponent and 6-bit significand).
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*
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* The relative error (e = r0*sqrt(m)-1) of a linear estimate
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* (r0 = a*m + b) is |e| < 0.085955 ~ 0x1.6p-4 at best;
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* a table lookup is faster and needs one less iteration.
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* The 7-bit table gives |e| < 0x1.fdp-9.
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*
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* A Newton-Raphson iteration for r is
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* s = m*r
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* d = s*r
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* u = 3 - d
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* r = r*u/2
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*
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* Fixed point representations:
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* m, s, d, u, three are all 2.30; r is 0.32
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*/
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m64 = a->frac_hi;
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m32 = m64 >> 32;
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r32 = rsqrt_tab[index] << 16;
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/* |r*sqrt(m) - 1| < 0x1.FDp-9 */
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s32 = ((uint64_t)m32 * r32) >> 32;
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d32 = ((uint64_t)s32 * r32) >> 32;
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u32 = three32 - d32;
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if (N == 64) {
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/* float64 or smaller */
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r32 = ((uint64_t)r32 * u32) >> 31;
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/* |r*sqrt(m) - 1| < 0x1.7Bp-16 */
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s32 = ((uint64_t)m32 * r32) >> 32;
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d32 = ((uint64_t)s32 * r32) >> 32;
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u32 = three32 - d32;
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if (fmt->frac_size <= 23) {
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/* float32 or smaller */
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s32 = ((uint64_t)s32 * u32) >> 32; /* 3.29 */
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s32 = (s32 - 1) >> 6; /* 9.23 */
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/* s < sqrt(m) < s + 0x1.08p-23 */
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/* compute nearest rounded result to 2.23 bits */
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uint32_t d0 = (m32 << 16) - s32 * s32;
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uint32_t d1 = s32 - d0;
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uint32_t d2 = d1 + s32 + 1;
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s32 += d1 >> 31;
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a->frac_hi = (uint64_t)s32 << (64 - 25);
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/* increment or decrement for inexact */
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if (d2 != 0) {
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a->frac_hi += ((int32_t)(d1 ^ d2) < 0 ? -1 : 1);
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}
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goto done;
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}
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/* float64 */
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r64 = (uint64_t)r32 * u32 * 2;
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/* |r*sqrt(m) - 1| < 0x1.37-p29; convert to 64-bit arithmetic */
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mul64To128(m64, r64, &s64, &discard);
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mul64To128(s64, r64, &d64, &discard);
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u64 = three64 - d64;
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mul64To128(s64, u64, &s64, &discard); /* 3.61 */
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s64 = (s64 - 2) >> 9; /* 12.52 */
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/* Compute nearest rounded result */
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uint64_t d0 = (m64 << 42) - s64 * s64;
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uint64_t d1 = s64 - d0;
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uint64_t d2 = d1 + s64 + 1;
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s64 += d1 >> 63;
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a->frac_hi = s64 << (64 - 54);
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/* increment or decrement for inexact */
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if (d2 != 0) {
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a->frac_hi += ((int64_t)(d1 ^ d2) < 0 ? -1 : 1);
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}
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goto done;
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}
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r64 = (uint64_t)r32 * u32 * 2;
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/* |r*sqrt(m) - 1| < 0x1.7Bp-16; convert to 64-bit arithmetic */
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mul64To128(m64, r64, &s64, &discard);
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mul64To128(s64, r64, &d64, &discard);
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u64 = three64 - d64;
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mul64To128(u64, r64, &r64, &discard);
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r64 <<= 1;
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/* |r*sqrt(m) - 1| < 0x1.a5p-31 */
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mul64To128(m64, r64, &s64, &discard);
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mul64To128(s64, r64, &d64, &discard);
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u64 = three64 - d64;
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mul64To128(u64, r64, &rh, &rl);
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add128(rh, rl, rh, rl, &rh, &rl);
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/* |r*sqrt(m) - 1| < 0x1.c001p-59; change to 128-bit arithmetic */
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mul128To256(a->frac_hi, a->frac_lo, rh, rl, &sh, &sl, &discard, &discard);
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mul128To256(sh, sl, rh, rl, &dh, &dl, &discard, &discard);
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sub128(three64, 0, dh, dl, &uh, &ul);
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mul128To256(uh, ul, sh, sl, &sh, &sl, &discard, &discard); /* 3.125 */
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/* -0x1p-116 < s - sqrt(m) < 0x3.8001p-125 */
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sub128(sh, sl, 0, 4, &sh, &sl);
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shift128Right(sh, sl, 13, &sh, &sl); /* 16.112 */
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/* s < sqrt(m) < s + 1ulp */
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/* Compute nearest rounded result */
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mul64To128(sl, sl, &d0h, &d0l);
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d0h += 2 * sh * sl;
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sub128(a->frac_lo << 34, 0, d0h, d0l, &d0h, &d0l);
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sub128(sh, sl, d0h, d0l, &d1h, &d1l);
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add128(sh, sl, 0, 1, &d2h, &d2l);
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add128(d2h, d2l, d1h, d1l, &d2h, &d2l);
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add128(sh, sl, 0, d1h >> 63, &sh, &sl);
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shift128Left(sh, sl, 128 - 114, &sh, &sl);
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/* increment or decrement for inexact */
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if (d2h | d2l) {
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if ((int64_t)(d1h ^ d2h) < 0) {
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sub128(sh, sl, 0, 1, &sh, &sl);
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} else {
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add128(sh, sl, 0, 1, &sh, &sl);
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}
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}
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a->frac_lo = sl;
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a->frac_hi = sh;
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done:
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/* Convert back from base 4 to base 2. */
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a->exp >>= 1;
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if (!(a->frac_hi & DECOMPOSED_IMPLICIT_BIT)) {
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frac_add(a, a, a);
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} else {
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a->exp += 1;
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}
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return;
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d_nan:
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float_raise(float_flag_invalid, status);
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parts_default_nan(a, status);
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}
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/*
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* Rounds the floating-point value `a' to an integer, and returns the
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* result as a floating-point value. The operation is performed
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207
fpu/softfloat.c
207
fpu/softfloat.c
@ -820,6 +820,12 @@ static FloatParts128 *parts128_div(FloatParts128 *a, FloatParts128 *b,
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#define parts_div(A, B, S) \
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PARTS_GENERIC_64_128(div, A)(A, B, S)
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static void parts64_sqrt(FloatParts64 *a, float_status *s, const FloatFmt *f);
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static void parts128_sqrt(FloatParts128 *a, float_status *s, const FloatFmt *f);
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#define parts_sqrt(A, S, F) \
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PARTS_GENERIC_64_128(sqrt, A)(A, S, F)
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static bool parts64_round_to_int_normal(FloatParts64 *a, FloatRoundMode rm,
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int scale, int frac_size);
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static bool parts128_round_to_int_normal(FloatParts128 *a, FloatRoundMode r,
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@ -1386,6 +1392,30 @@ static void frac128_widen(FloatParts256 *r, FloatParts128 *a)
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#define frac_widen(A, B) FRAC_GENERIC_64_128(widen, B)(A, B)
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/*
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* Reciprocal sqrt table. 1 bit of exponent, 6-bits of mantessa.
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* From https://git.musl-libc.org/cgit/musl/tree/src/math/sqrt_data.c
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* and thus MIT licenced.
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*/
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static const uint16_t rsqrt_tab[128] = {
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0xb451, 0xb2f0, 0xb196, 0xb044, 0xaef9, 0xadb6, 0xac79, 0xab43,
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0xaa14, 0xa8eb, 0xa7c8, 0xa6aa, 0xa592, 0xa480, 0xa373, 0xa26b,
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0xa168, 0xa06a, 0x9f70, 0x9e7b, 0x9d8a, 0x9c9d, 0x9bb5, 0x9ad1,
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0x99f0, 0x9913, 0x983a, 0x9765, 0x9693, 0x95c4, 0x94f8, 0x9430,
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0x936b, 0x92a9, 0x91ea, 0x912e, 0x9075, 0x8fbe, 0x8f0a, 0x8e59,
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0x8daa, 0x8cfe, 0x8c54, 0x8bac, 0x8b07, 0x8a64, 0x89c4, 0x8925,
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0x8889, 0x87ee, 0x8756, 0x86c0, 0x862b, 0x8599, 0x8508, 0x8479,
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0x83ec, 0x8361, 0x82d8, 0x8250, 0x81c9, 0x8145, 0x80c2, 0x8040,
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0xff02, 0xfd0e, 0xfb25, 0xf947, 0xf773, 0xf5aa, 0xf3ea, 0xf234,
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0xf087, 0xeee3, 0xed47, 0xebb3, 0xea27, 0xe8a3, 0xe727, 0xe5b2,
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0xe443, 0xe2dc, 0xe17a, 0xe020, 0xdecb, 0xdd7d, 0xdc34, 0xdaf1,
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0xd9b3, 0xd87b, 0xd748, 0xd61a, 0xd4f1, 0xd3cd, 0xd2ad, 0xd192,
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0xd07b, 0xcf69, 0xce5b, 0xcd51, 0xcc4a, 0xcb48, 0xca4a, 0xc94f,
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0xc858, 0xc764, 0xc674, 0xc587, 0xc49d, 0xc3b7, 0xc2d4, 0xc1f4,
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0xc116, 0xc03c, 0xbf65, 0xbe90, 0xbdbe, 0xbcef, 0xbc23, 0xbb59,
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0xba91, 0xb9cc, 0xb90a, 0xb84a, 0xb78c, 0xb6d0, 0xb617, 0xb560,
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};
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#define partsN(NAME) glue(glue(glue(parts,N),_),NAME)
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#define FloatPartsN glue(FloatParts,N)
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#define FloatPartsW glue(FloatParts,W)
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@ -3586,103 +3616,35 @@ float128 float128_scalbn(float128 a, int n, float_status *status)
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/*
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* Square Root
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*
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* The old softfloat code did an approximation step before zeroing in
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* on the final result. However for simpleness we just compute the
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* square root by iterating down from the implicit bit to enough extra
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* bits to ensure we get a correctly rounded result.
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*
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* This does mean however the calculation is slower than before,
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* especially for 64 bit floats.
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*/
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static FloatParts64 sqrt_float(FloatParts64 a, float_status *s, const FloatFmt *p)
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{
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uint64_t a_frac, r_frac, s_frac;
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int bit, last_bit;
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if (is_nan(a.cls)) {
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parts_return_nan(&a, s);
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return a;
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}
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if (a.cls == float_class_zero) {
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return a; /* sqrt(+-0) = +-0 */
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}
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if (a.sign) {
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float_raise(float_flag_invalid, s);
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parts_default_nan(&a, s);
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return a;
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}
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if (a.cls == float_class_inf) {
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return a; /* sqrt(+inf) = +inf */
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}
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assert(a.cls == float_class_normal);
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/* We need two overflow bits at the top. Adding room for that is a
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* right shift. If the exponent is odd, we can discard the low bit
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* by multiplying the fraction by 2; that's a left shift. Combine
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* those and we shift right by 1 if the exponent is odd, otherwise 2.
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*/
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a_frac = a.frac >> (2 - (a.exp & 1));
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a.exp >>= 1;
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/* Bit-by-bit computation of sqrt. */
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r_frac = 0;
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s_frac = 0;
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/* Iterate from implicit bit down to the 3 extra bits to compute a
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* properly rounded result. Remember we've inserted two more bits
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* at the top, so these positions are two less.
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*/
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bit = DECOMPOSED_BINARY_POINT - 2;
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last_bit = MAX(p->frac_shift - 4, 0);
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do {
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uint64_t q = 1ULL << bit;
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uint64_t t_frac = s_frac + q;
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if (t_frac <= a_frac) {
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s_frac = t_frac + q;
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a_frac -= t_frac;
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r_frac += q;
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}
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a_frac <<= 1;
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} while (--bit >= last_bit);
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/* Undo the right shift done above. If there is any remaining
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* fraction, the result is inexact. Set the sticky bit.
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*/
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a.frac = (r_frac << 2) + (a_frac != 0);
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return a;
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}
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float16 QEMU_FLATTEN float16_sqrt(float16 a, float_status *status)
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{
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FloatParts64 pa, pr;
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FloatParts64 p;
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float16_unpack_canonical(&pa, a, status);
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pr = sqrt_float(pa, status, &float16_params);
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return float16_round_pack_canonical(&pr, status);
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float16_unpack_canonical(&p, a, status);
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parts_sqrt(&p, status, &float16_params);
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return float16_round_pack_canonical(&p, status);
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}
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static float32 QEMU_SOFTFLOAT_ATTR
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soft_f32_sqrt(float32 a, float_status *status)
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{
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FloatParts64 pa, pr;
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FloatParts64 p;
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float32_unpack_canonical(&pa, a, status);
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pr = sqrt_float(pa, status, &float32_params);
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return float32_round_pack_canonical(&pr, status);
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float32_unpack_canonical(&p, a, status);
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parts_sqrt(&p, status, &float32_params);
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return float32_round_pack_canonical(&p, status);
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}
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static float64 QEMU_SOFTFLOAT_ATTR
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soft_f64_sqrt(float64 a, float_status *status)
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{
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FloatParts64 pa, pr;
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FloatParts64 p;
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float64_unpack_canonical(&pa, a, status);
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pr = sqrt_float(pa, status, &float64_params);
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return float64_round_pack_canonical(&pr, status);
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float64_unpack_canonical(&p, a, status);
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parts_sqrt(&p, status, &float64_params);
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return float64_round_pack_canonical(&p, status);
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}
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float32 QEMU_FLATTEN float32_sqrt(float32 xa, float_status *s)
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@ -3741,11 +3703,20 @@ float64 QEMU_FLATTEN float64_sqrt(float64 xa, float_status *s)
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bfloat16 QEMU_FLATTEN bfloat16_sqrt(bfloat16 a, float_status *status)
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{
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FloatParts64 pa, pr;
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FloatParts64 p;
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bfloat16_unpack_canonical(&pa, a, status);
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pr = sqrt_float(pa, status, &bfloat16_params);
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return bfloat16_round_pack_canonical(&pr, status);
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bfloat16_unpack_canonical(&p, a, status);
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parts_sqrt(&p, status, &bfloat16_params);
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return bfloat16_round_pack_canonical(&p, status);
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}
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float128 QEMU_FLATTEN float128_sqrt(float128 a, float_status *status)
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{
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FloatParts128 p;
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|
||||
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)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user