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qemu/fpu/softfloat-parts.c.inc
Richard Henderson 722460652b fpu: Handle m68k extended precision denormals properly 
Motorola treats denormals with explicit integer bit set as
having unbiased exponent 0, unlike Intel which treats it as
having unbiased exponent 1 (more like all other IEEE formats
that have no explicit integer bit).

Add a flag on FloatFmt to differentiate the behaviour.

Reported-by: Keith Packard <keithp@keithp.com>
Reviewed-by: Philippe Mathieu-Daudé <philmd@linaro.org>
Signed-off-by: Richard Henderson <richard.henderson@linaro.org>
2023-09-16 14:57:16 +00:00

1626 lines
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/*
* QEMU float support
*
* The code in this source file is derived from release 2a of the SoftFloat
* IEC/IEEE Floating-point Arithmetic Package. Those parts of the code (and
* some later contributions) are provided under that license, as detailed below.
* It has subsequently been modified by contributors to the QEMU Project,
* so some portions are provided under:
* the SoftFloat-2a license
* the BSD license
* GPL-v2-or-later
*
* Any future contributions to this file after December 1st 2014 will be
* taken to be licensed under the Softfloat-2a license unless specifically
* indicated otherwise.
*/
static void partsN(return_nan)(FloatPartsN *a, float_status *s)
{
switch (a->cls) {
case float_class_snan:
float_raise(float_flag_invalid | float_flag_invalid_snan, s);
if (s->default_nan_mode) {
parts_default_nan(a, s);
} else {
parts_silence_nan(a, s);
}
break;
case float_class_qnan:
if (s->default_nan_mode) {
parts_default_nan(a, s);
}
break;
default:
g_assert_not_reached();
}
}
static FloatPartsN *partsN(pick_nan)(FloatPartsN *a, FloatPartsN *b,
float_status *s)
{
if (is_snan(a->cls) || is_snan(b->cls)) {
float_raise(float_flag_invalid | float_flag_invalid_snan, s);
}
if (s->default_nan_mode) {
parts_default_nan(a, s);
} else {
int cmp = frac_cmp(a, b);
if (cmp == 0) {
cmp = a->sign < b->sign;
}
if (pickNaN(a->cls, b->cls, cmp > 0, s)) {
a = b;
}
if (is_snan(a->cls)) {
parts_silence_nan(a, s);
}
}
return a;
}
static FloatPartsN *partsN(pick_nan_muladd)(FloatPartsN *a, FloatPartsN *b,
FloatPartsN *c, float_status *s,
int ab_mask, int abc_mask)
{
int which;
if (unlikely(abc_mask & float_cmask_snan)) {
float_raise(float_flag_invalid | float_flag_invalid_snan, s);
}
which = pickNaNMulAdd(a->cls, b->cls, c->cls,
ab_mask == float_cmask_infzero, s);
if (s->default_nan_mode || which == 3) {
/*
* Note that this check is after pickNaNMulAdd so that function
* has an opportunity to set the Invalid flag for infzero.
*/
parts_default_nan(a, s);
return a;
}
switch (which) {
case 0:
break;
case 1:
a = b;
break;
case 2:
a = c;
break;
default:
g_assert_not_reached();
}
if (is_snan(a->cls)) {
parts_silence_nan(a, s);
}
return a;
}
/*
* Canonicalize the FloatParts structure. Determine the class,
* unbias the exponent, and normalize the fraction.
*/
static void partsN(canonicalize)(FloatPartsN *p, float_status *status,
const FloatFmt *fmt)
{
if (unlikely(p->exp == 0)) {
if (likely(frac_eqz(p))) {
p->cls = float_class_zero;
} else if (status->flush_inputs_to_zero) {
float_raise(float_flag_input_denormal, status);
p->cls = float_class_zero;
frac_clear(p);
} else {
int shift = frac_normalize(p);
p->cls = float_class_normal;
p->exp = fmt->frac_shift - fmt->exp_bias
- shift + !fmt->m68k_denormal;
}
} else if (likely(p->exp < fmt->exp_max) || fmt->arm_althp) {
p->cls = float_class_normal;
p->exp -= fmt->exp_bias;
frac_shl(p, fmt->frac_shift);
p->frac_hi |= DECOMPOSED_IMPLICIT_BIT;
} else if (likely(frac_eqz(p))) {
p->cls = float_class_inf;
} else {
frac_shl(p, fmt->frac_shift);
p->cls = (parts_is_snan_frac(p->frac_hi, status)
? float_class_snan : float_class_qnan);
}
}
/*
* Round and uncanonicalize a floating-point number by parts. There
* are FRAC_SHIFT bits that may require rounding at the bottom of the
* fraction; these bits will be removed. The exponent will be biased
* by EXP_BIAS and must be bounded by [EXP_MAX-1, 0].
*/
static void partsN(uncanon_normal)(FloatPartsN *p, float_status *s,
const FloatFmt *fmt)
{
const int exp_max = fmt->exp_max;
const int frac_shift = fmt->frac_shift;
const uint64_t round_mask = fmt->round_mask;
const uint64_t frac_lsb = round_mask + 1;
const uint64_t frac_lsbm1 = round_mask ^ (round_mask >> 1);
const uint64_t roundeven_mask = round_mask | frac_lsb;
uint64_t inc;
bool overflow_norm = false;
int exp, flags = 0;
switch (s->float_rounding_mode) {
case float_round_nearest_even:
if (N > 64 && frac_lsb == 0) {
inc = ((p->frac_hi & 1) || (p->frac_lo & round_mask) != frac_lsbm1
? frac_lsbm1 : 0);
} else {
inc = ((p->frac_lo & roundeven_mask) != frac_lsbm1
? frac_lsbm1 : 0);
}
break;
case float_round_ties_away:
inc = frac_lsbm1;
break;
case float_round_to_zero:
overflow_norm = true;
inc = 0;
break;
case float_round_up:
inc = p->sign ? 0 : round_mask;
overflow_norm = p->sign;
break;
case float_round_down:
inc = p->sign ? round_mask : 0;
overflow_norm = !p->sign;
break;
case float_round_to_odd:
overflow_norm = true;
/* fall through */
case float_round_to_odd_inf:
if (N > 64 && frac_lsb == 0) {
inc = p->frac_hi & 1 ? 0 : round_mask;
} else {
inc = p->frac_lo & frac_lsb ? 0 : round_mask;
}
break;
default:
g_assert_not_reached();
}
exp = p->exp + fmt->exp_bias;
if (likely(exp > 0)) {
if (p->frac_lo & round_mask) {
flags |= float_flag_inexact;
if (frac_addi(p, p, inc)) {
frac_shr(p, 1);
p->frac_hi |= DECOMPOSED_IMPLICIT_BIT;
exp++;
}
p->frac_lo &= ~round_mask;
}
if (fmt->arm_althp) {
/* ARM Alt HP eschews Inf and NaN for a wider exponent. */
if (unlikely(exp > exp_max)) {
/* Overflow. Return the maximum normal. */
flags = float_flag_invalid;
exp = exp_max;
frac_allones(p);
p->frac_lo &= ~round_mask;
}
} else if (unlikely(exp >= exp_max)) {
flags |= float_flag_overflow;
if (s->rebias_overflow) {
exp -= fmt->exp_re_bias;
} else if (overflow_norm) {
flags |= float_flag_inexact;
exp = exp_max - 1;
frac_allones(p);
p->frac_lo &= ~round_mask;
} else {
flags |= float_flag_inexact;
p->cls = float_class_inf;
exp = exp_max;
frac_clear(p);
}
}
frac_shr(p, frac_shift);
} else if (unlikely(s->rebias_underflow)) {
flags |= float_flag_underflow;
exp += fmt->exp_re_bias;
if (p->frac_lo & round_mask) {
flags |= float_flag_inexact;
if (frac_addi(p, p, inc)) {
frac_shr(p, 1);
p->frac_hi |= DECOMPOSED_IMPLICIT_BIT;
exp++;
}
p->frac_lo &= ~round_mask;
}
frac_shr(p, frac_shift);
} else if (s->flush_to_zero) {
flags |= float_flag_output_denormal;
p->cls = float_class_zero;
exp = 0;
frac_clear(p);
} else {
bool is_tiny = s->tininess_before_rounding || exp < 0;
if (!is_tiny) {
FloatPartsN discard;
is_tiny = !frac_addi(&discard, p, inc);
}
frac_shrjam(p, !fmt->m68k_denormal - exp);
if (p->frac_lo & round_mask) {
/* Need to recompute round-to-even/round-to-odd. */
switch (s->float_rounding_mode) {
case float_round_nearest_even:
if (N > 64 && frac_lsb == 0) {
inc = ((p->frac_hi & 1) ||
(p->frac_lo & round_mask) != frac_lsbm1
? frac_lsbm1 : 0);
} else {
inc = ((p->frac_lo & roundeven_mask) != frac_lsbm1
? frac_lsbm1 : 0);
}
break;
case float_round_to_odd:
case float_round_to_odd_inf:
if (N > 64 && frac_lsb == 0) {
inc = p->frac_hi & 1 ? 0 : round_mask;
} else {
inc = p->frac_lo & frac_lsb ? 0 : round_mask;
}
break;
default:
break;
}
flags |= float_flag_inexact;
frac_addi(p, p, inc);
p->frac_lo &= ~round_mask;
}
exp = (p->frac_hi & DECOMPOSED_IMPLICIT_BIT) && !fmt->m68k_denormal;
frac_shr(p, frac_shift);
if (is_tiny && (flags & float_flag_inexact)) {
flags |= float_flag_underflow;
}
if (exp == 0 && frac_eqz(p)) {
p->cls = float_class_zero;
}
}
p->exp = exp;
float_raise(flags, s);
}
static void partsN(uncanon)(FloatPartsN *p, float_status *s,
const FloatFmt *fmt)
{
if (likely(p->cls == float_class_normal)) {
parts_uncanon_normal(p, s, fmt);
} else {
switch (p->cls) {
case float_class_zero:
p->exp = 0;
frac_clear(p);
return;
case float_class_inf:
g_assert(!fmt->arm_althp);
p->exp = fmt->exp_max;
frac_clear(p);
return;
case float_class_qnan:
case float_class_snan:
g_assert(!fmt->arm_althp);
p->exp = fmt->exp_max;
frac_shr(p, fmt->frac_shift);
return;
default:
break;
}
g_assert_not_reached();
}
}
/*
* Returns the result of adding or subtracting the values of the
* floating-point values `a' and `b'. The operation is performed
* according to the IEC/IEEE Standard for Binary Floating-Point
* Arithmetic.
*/
static FloatPartsN *partsN(addsub)(FloatPartsN *a, FloatPartsN *b,
float_status *s, bool subtract)
{
bool b_sign = b->sign ^ subtract;
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
if (a->sign != b_sign) {
/* Subtraction */
if (likely(ab_mask == float_cmask_normal)) {
if (parts_sub_normal(a, b)) {
return a;
}
/* Subtract was exact, fall through to set sign. */
ab_mask = float_cmask_zero;
}
if (ab_mask == float_cmask_zero) {
a->sign = s->float_rounding_mode == float_round_down;
return a;
}
if (unlikely(ab_mask & float_cmask_anynan)) {
goto p_nan;
}
if (ab_mask & float_cmask_inf) {
if (a->cls != float_class_inf) {
/* N - Inf */
goto return_b;
}
if (b->cls != float_class_inf) {
/* Inf - N */
return a;
}
/* Inf - Inf */
float_raise(float_flag_invalid | float_flag_invalid_isi, s);
parts_default_nan(a, s);
return a;
}
} else {
/* Addition */
if (likely(ab_mask == float_cmask_normal)) {
parts_add_normal(a, b);
return a;
}
if (ab_mask == float_cmask_zero) {
return a;
}
if (unlikely(ab_mask & float_cmask_anynan)) {
goto p_nan;
}
if (ab_mask & float_cmask_inf) {
a->cls = float_class_inf;
return a;
}
}
if (b->cls == float_class_zero) {
g_assert(a->cls == float_class_normal);
return a;
}
g_assert(a->cls == float_class_zero);
g_assert(b->cls == float_class_normal);
return_b:
b->sign = b_sign;
return b;
p_nan:
return parts_pick_nan(a, b, s);
}
/*
* Returns the result of multiplying the floating-point values `a' and
* `b'. The operation is performed according to the IEC/IEEE Standard
* for Binary Floating-Point Arithmetic.
*/
static FloatPartsN *partsN(mul)(FloatPartsN *a, FloatPartsN *b,
float_status *s)
{
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
bool sign = a->sign ^ b->sign;
if (likely(ab_mask == float_cmask_normal)) {
FloatPartsW tmp;
frac_mulw(&tmp, a, b);
frac_truncjam(a, &tmp);
a->exp += b->exp + 1;
if (!(a->frac_hi & DECOMPOSED_IMPLICIT_BIT)) {
frac_add(a, a, a);
a->exp -= 1;
}
a->sign = sign;
return a;
}
/* Inf * Zero == NaN */
if (unlikely(ab_mask == float_cmask_infzero)) {
float_raise(float_flag_invalid | float_flag_invalid_imz, s);
parts_default_nan(a, s);
return a;
}
if (unlikely(ab_mask & float_cmask_anynan)) {
return parts_pick_nan(a, b, s);
}
/* Multiply by 0 or Inf */
if (ab_mask & float_cmask_inf) {
a->cls = float_class_inf;
a->sign = sign;
return a;
}
g_assert(ab_mask & float_cmask_zero);
a->cls = float_class_zero;
a->sign = sign;
return a;
}
/*
* Returns the result of multiplying the floating-point values `a' and
* `b' then adding 'c', with no intermediate rounding step after the
* multiplication. The operation is performed according to the
* IEC/IEEE Standard for Binary Floating-Point Arithmetic 754-2008.
* The flags argument allows the caller to select negation of the
* addend, the intermediate product, or the final result. (The
* difference between this and having the caller do a separate
* negation is that negating externally will flip the sign bit on NaNs.)
*
* Requires A and C extracted into a double-sized structure to provide the
* extra space for the widening multiply.
*/
static FloatPartsN *partsN(muladd)(FloatPartsN *a, FloatPartsN *b,
FloatPartsN *c, int flags, float_status *s)
{
int ab_mask, abc_mask;
FloatPartsW p_widen, c_widen;
ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
abc_mask = float_cmask(c->cls) | ab_mask;
/*
* It is implementation-defined whether the cases of (0,inf,qnan)
* and (inf,0,qnan) raise InvalidOperation or not (and what QNaN
* they return if they do), so we have to hand this information
* off to the target-specific pick-a-NaN routine.
*/
if (unlikely(abc_mask & float_cmask_anynan)) {
return parts_pick_nan_muladd(a, b, c, s, ab_mask, abc_mask);
}
if (flags & float_muladd_negate_c) {
c->sign ^= 1;
}
/* Compute the sign of the product into A. */
a->sign ^= b->sign;
if (flags & float_muladd_negate_product) {
a->sign ^= 1;
}
if (unlikely(ab_mask != float_cmask_normal)) {
if (unlikely(ab_mask == float_cmask_infzero)) {
float_raise(float_flag_invalid | float_flag_invalid_imz, s);
goto d_nan;
}
if (ab_mask & float_cmask_inf) {
if (c->cls == float_class_inf && a->sign != c->sign) {
float_raise(float_flag_invalid | float_flag_invalid_isi, s);
goto d_nan;
}
goto return_inf;
}
g_assert(ab_mask & float_cmask_zero);
if (c->cls == float_class_normal) {
*a = *c;
goto return_normal;
}
if (c->cls == float_class_zero) {
if (a->sign != c->sign) {
goto return_sub_zero;
}
goto return_zero;
}
g_assert(c->cls == float_class_inf);
}
if (unlikely(c->cls == float_class_inf)) {
a->sign = c->sign;
goto return_inf;
}
/* Perform the multiplication step. */
p_widen.sign = a->sign;
p_widen.exp = a->exp + b->exp + 1;
frac_mulw(&p_widen, a, b);
if (!(p_widen.frac_hi & DECOMPOSED_IMPLICIT_BIT)) {
frac_add(&p_widen, &p_widen, &p_widen);
p_widen.exp -= 1;
}
/* Perform the addition step. */
if (c->cls != float_class_zero) {
/* Zero-extend C to less significant bits. */
frac_widen(&c_widen, c);
c_widen.exp = c->exp;
if (a->sign == c->sign) {
parts_add_normal(&p_widen, &c_widen);
} else if (!parts_sub_normal(&p_widen, &c_widen)) {
goto return_sub_zero;
}
}
/* Narrow with sticky bit, for proper rounding later. */
frac_truncjam(a, &p_widen);
a->sign = p_widen.sign;
a->exp = p_widen.exp;
return_normal:
if (flags & float_muladd_halve_result) {
a->exp -= 1;
}
finish_sign:
if (flags & float_muladd_negate_result) {
a->sign ^= 1;
}
return a;
return_sub_zero:
a->sign = s->float_rounding_mode == float_round_down;
return_zero:
a->cls = float_class_zero;
goto finish_sign;
return_inf:
a->cls = float_class_inf;
goto finish_sign;
d_nan:
parts_default_nan(a, s);
return a;
}
/*
* Returns the result of dividing the floating-point value `a' by the
* corresponding value `b'. The operation is performed according to
* the IEC/IEEE Standard for Binary Floating-Point Arithmetic.
*/
static FloatPartsN *partsN(div)(FloatPartsN *a, FloatPartsN *b,
float_status *s)
{
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
bool sign = a->sign ^ b->sign;
if (likely(ab_mask == float_cmask_normal)) {
a->sign = sign;
a->exp -= b->exp + frac_div(a, b);
return a;
}
/* 0/0 or Inf/Inf => NaN */
if (unlikely(ab_mask == float_cmask_zero)) {
float_raise(float_flag_invalid | float_flag_invalid_zdz, s);
goto d_nan;
}
if (unlikely(ab_mask == float_cmask_inf)) {
float_raise(float_flag_invalid | float_flag_invalid_idi, s);
goto d_nan;
}
/* All the NaN cases */
if (unlikely(ab_mask & float_cmask_anynan)) {
return parts_pick_nan(a, b, s);
}
a->sign = sign;
/* Inf / X */
if (a->cls == float_class_inf) {
return a;
}
/* 0 / X */
if (a->cls == float_class_zero) {
return a;
}
/* X / Inf */
if (b->cls == float_class_inf) {
a->cls = float_class_zero;
return a;
}
/* X / 0 => Inf */
g_assert(b->cls == float_class_zero);
float_raise(float_flag_divbyzero, s);
a->cls = float_class_inf;
return a;
d_nan:
parts_default_nan(a, s);
return a;
}
/*
* Floating point remainder, per IEC/IEEE, or modulus.
*/
static FloatPartsN *partsN(modrem)(FloatPartsN *a, FloatPartsN *b,
uint64_t *mod_quot, float_status *s)
{
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
if (likely(ab_mask == float_cmask_normal)) {
frac_modrem(a, b, mod_quot);
return a;
}
if (mod_quot) {
*mod_quot = 0;
}
/* All the NaN cases */
if (unlikely(ab_mask & float_cmask_anynan)) {
return parts_pick_nan(a, b, s);
}
/* Inf % N; N % 0 */
if (a->cls == float_class_inf || b->cls == float_class_zero) {
float_raise(float_flag_invalid, s);
parts_default_nan(a, s);
return a;
}
/* N % Inf; 0 % N */
g_assert(b->cls == float_class_inf || a->cls == float_class_zero);
return a;
}
/*
* 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 | float_flag_invalid_sqrt, 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
* Arithmetic.
*
* parts_round_to_int_normal is an internal helper function for
* normal numbers only, returning true for inexact but not directly
* raising float_flag_inexact.
*/
static bool partsN(round_to_int_normal)(FloatPartsN *a, FloatRoundMode rmode,
int scale, int frac_size)
{
uint64_t frac_lsb, frac_lsbm1, rnd_even_mask, rnd_mask, inc;
int shift_adj;
scale = MIN(MAX(scale, -0x10000), 0x10000);
a->exp += scale;
if (a->exp < 0) {
bool one;
/* All fractional */
switch (rmode) {
case float_round_nearest_even:
one = false;
if (a->exp == -1) {
FloatPartsN tmp;
/* Shift left one, discarding DECOMPOSED_IMPLICIT_BIT */
frac_add(&tmp, a, a);
/* Anything remaining means frac > 0.5. */
one = !frac_eqz(&tmp);
}
break;
case float_round_ties_away:
one = a->exp == -1;
break;
case float_round_to_zero:
one = false;
break;
case float_round_up:
one = !a->sign;
break;
case float_round_down:
one = a->sign;
break;
case float_round_to_odd:
one = true;
break;
default:
g_assert_not_reached();
}
frac_clear(a);
a->exp = 0;
if (one) {
a->frac_hi = DECOMPOSED_IMPLICIT_BIT;
} else {
a->cls = float_class_zero;
}
return true;
}
if (a->exp >= frac_size) {
/* All integral */
return false;
}
if (N > 64 && a->exp < N - 64) {
/*
* Rounding is not in the low word -- shift lsb to bit 2,
* which leaves room for sticky and rounding bit.
*/
shift_adj = (N - 1) - (a->exp + 2);
frac_shrjam(a, shift_adj);
frac_lsb = 1 << 2;
} else {
shift_adj = 0;
frac_lsb = DECOMPOSED_IMPLICIT_BIT >> (a->exp & 63);
}
frac_lsbm1 = frac_lsb >> 1;
rnd_mask = frac_lsb - 1;
rnd_even_mask = rnd_mask | frac_lsb;
if (!(a->frac_lo & rnd_mask)) {
/* Fractional bits already clear, undo the shift above. */
frac_shl(a, shift_adj);
return false;
}
switch (rmode) {
case float_round_nearest_even:
inc = ((a->frac_lo & rnd_even_mask) != frac_lsbm1 ? frac_lsbm1 : 0);
break;
case float_round_ties_away:
inc = frac_lsbm1;
break;
case float_round_to_zero:
inc = 0;
break;
case float_round_up:
inc = a->sign ? 0 : rnd_mask;
break;
case float_round_down:
inc = a->sign ? rnd_mask : 0;
break;
case float_round_to_odd:
inc = a->frac_lo & frac_lsb ? 0 : rnd_mask;
break;
default:
g_assert_not_reached();
}
if (shift_adj == 0) {
if (frac_addi(a, a, inc)) {
frac_shr(a, 1);
a->frac_hi |= DECOMPOSED_IMPLICIT_BIT;
a->exp++;
}
a->frac_lo &= ~rnd_mask;
} else {
frac_addi(a, a, inc);
a->frac_lo &= ~rnd_mask;
/* Be careful shifting back, not to overflow */
frac_shl(a, shift_adj - 1);
if (a->frac_hi & DECOMPOSED_IMPLICIT_BIT) {
a->exp++;
} else {
frac_add(a, a, a);
}
}
return true;
}
static void partsN(round_to_int)(FloatPartsN *a, FloatRoundMode rmode,
int scale, float_status *s,
const FloatFmt *fmt)
{
switch (a->cls) {
case float_class_qnan:
case float_class_snan:
parts_return_nan(a, s);
break;
case float_class_zero:
case float_class_inf:
break;
case float_class_normal:
if (parts_round_to_int_normal(a, rmode, scale, fmt->frac_size)) {
float_raise(float_flag_inexact, s);
}
break;
default:
g_assert_not_reached();
}
}
/*
* Returns the result of converting the floating-point value `a' to
* the two's complement integer format. The conversion is performed
* according to the IEC/IEEE Standard for Binary Floating-Point
* Arithmetic---which means in particular that the conversion is
* rounded according to the current rounding mode. If `a' is a NaN,
* the largest positive integer is returned. Otherwise, if the
* conversion overflows, the largest integer with the same sign as `a'
* is returned.
*/
static int64_t partsN(float_to_sint)(FloatPartsN *p, FloatRoundMode rmode,
int scale, int64_t min, int64_t max,
float_status *s)
{
int flags = 0;
uint64_t r;
switch (p->cls) {
case float_class_snan:
flags |= float_flag_invalid_snan;
/* fall through */
case float_class_qnan:
flags |= float_flag_invalid;
r = max;
break;
case float_class_inf:
flags = float_flag_invalid | float_flag_invalid_cvti;
r = p->sign ? min : max;
break;
case float_class_zero:
return 0;
case float_class_normal:
/* TODO: N - 2 is frac_size for rounding; could use input fmt. */
if (parts_round_to_int_normal(p, rmode, scale, N - 2)) {
flags = float_flag_inexact;
}
if (p->exp <= DECOMPOSED_BINARY_POINT) {
r = p->frac_hi >> (DECOMPOSED_BINARY_POINT - p->exp);
} else {
r = UINT64_MAX;
}
if (p->sign) {
if (r <= -(uint64_t)min) {
r = -r;
} else {
flags = float_flag_invalid | float_flag_invalid_cvti;
r = min;
}
} else if (r > max) {
flags = float_flag_invalid | float_flag_invalid_cvti;
r = max;
}
break;
default:
g_assert_not_reached();
}
float_raise(flags, s);
return r;
}
/*
* Returns the result of converting the floating-point value `a' to
* the unsigned integer format. The conversion is performed according
* to the IEC/IEEE Standard for Binary Floating-Point
* Arithmetic---which means in particular that the conversion is
* rounded according to the current rounding mode. If `a' is a NaN,
* the largest unsigned integer is returned. Otherwise, if the
* conversion overflows, the largest unsigned integer is returned. If
* the 'a' is negative, the result is rounded and zero is returned;
* values that do not round to zero will raise the inexact exception
* flag.
*/
static uint64_t partsN(float_to_uint)(FloatPartsN *p, FloatRoundMode rmode,
int scale, uint64_t max, float_status *s)
{
int flags = 0;
uint64_t r;
switch (p->cls) {
case float_class_snan:
flags |= float_flag_invalid_snan;
/* fall through */
case float_class_qnan:
flags |= float_flag_invalid;
r = max;
break;
case float_class_inf:
flags = float_flag_invalid | float_flag_invalid_cvti;
r = p->sign ? 0 : max;
break;
case float_class_zero:
return 0;
case float_class_normal:
/* TODO: N - 2 is frac_size for rounding; could use input fmt. */
if (parts_round_to_int_normal(p, rmode, scale, N - 2)) {
flags = float_flag_inexact;
if (p->cls == float_class_zero) {
r = 0;
break;
}
}
if (p->sign) {
flags = float_flag_invalid | float_flag_invalid_cvti;
r = 0;
} else if (p->exp > DECOMPOSED_BINARY_POINT) {
flags = float_flag_invalid | float_flag_invalid_cvti;
r = max;
} else {
r = p->frac_hi >> (DECOMPOSED_BINARY_POINT - p->exp);
if (r > max) {
flags = float_flag_invalid | float_flag_invalid_cvti;
r = max;
}
}
break;
default:
g_assert_not_reached();
}
float_raise(flags, s);
return r;
}
/*
* Like partsN(float_to_sint), except do not saturate the result.
* Instead, return the rounded unbounded precision two's compliment result,
* modulo 2**(bitsm1 + 1).
*/
static int64_t partsN(float_to_sint_modulo)(FloatPartsN *p,
FloatRoundMode rmode,
int bitsm1, float_status *s)
{
int flags = 0;
uint64_t r;
bool overflow = false;
switch (p->cls) {
case float_class_snan:
flags |= float_flag_invalid_snan;
/* fall through */
case float_class_qnan:
flags |= float_flag_invalid;
r = 0;
break;
case float_class_inf:
overflow = true;
r = 0;
break;
case float_class_zero:
return 0;
case float_class_normal:
/* TODO: N - 2 is frac_size for rounding; could use input fmt. */
if (parts_round_to_int_normal(p, rmode, 0, N - 2)) {
flags = float_flag_inexact;
}
if (p->exp <= DECOMPOSED_BINARY_POINT) {
/*
* Because we rounded to integral, and exp < 64,
* we know frac_low is zero.
*/
r = p->frac_hi >> (DECOMPOSED_BINARY_POINT - p->exp);
if (p->exp < bitsm1) {
/* Result in range. */
} else if (p->exp == bitsm1) {
/* The only in-range value is INT_MIN. */
overflow = !p->sign || p->frac_hi != DECOMPOSED_IMPLICIT_BIT;
} else {
overflow = true;
}
} else {
/* Overflow, but there might still be bits to return. */
int shl = p->exp - DECOMPOSED_BINARY_POINT;
if (shl < N) {
frac_shl(p, shl);
r = p->frac_hi;
} else {
r = 0;
}
overflow = true;
}
if (p->sign) {
r = -r;
}
break;
default:
g_assert_not_reached();
}
if (overflow) {
flags = float_flag_invalid | float_flag_invalid_cvti;
}
float_raise(flags, s);
return r;
}
/*
* Integer to float conversions
*
* Returns the result of converting the two's complement integer `a'
* to the floating-point format. The conversion is performed according
* to the IEC/IEEE Standard for Binary Floating-Point Arithmetic.
*/
static void partsN(sint_to_float)(FloatPartsN *p, int64_t a,
int scale, float_status *s)
{
uint64_t f = a;
int shift;
memset(p, 0, sizeof(*p));
if (a == 0) {
p->cls = float_class_zero;
return;
}
p->cls = float_class_normal;
if (a < 0) {
f = -f;
p->sign = true;
}
shift = clz64(f);
scale = MIN(MAX(scale, -0x10000), 0x10000);
p->exp = DECOMPOSED_BINARY_POINT - shift + scale;
p->frac_hi = f << shift;
}
/*
* Unsigned Integer to float conversions
*
* Returns the result of converting the unsigned integer `a' to the
* floating-point format. The conversion is performed according to the
* IEC/IEEE Standard for Binary Floating-Point Arithmetic.
*/
static void partsN(uint_to_float)(FloatPartsN *p, uint64_t a,
int scale, float_status *status)
{
memset(p, 0, sizeof(*p));
if (a == 0) {
p->cls = float_class_zero;
} else {
int shift = clz64(a);
scale = MIN(MAX(scale, -0x10000), 0x10000);
p->cls = float_class_normal;
p->exp = DECOMPOSED_BINARY_POINT - shift + scale;
p->frac_hi = a << shift;
}
}
/*
* Float min/max.
*/
static FloatPartsN *partsN(minmax)(FloatPartsN *a, FloatPartsN *b,
float_status *s, int flags)
{
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
int a_exp, b_exp, cmp;
if (unlikely(ab_mask & float_cmask_anynan)) {
/*
* For minNum/maxNum (IEEE 754-2008)
* or minimumNumber/maximumNumber (IEEE 754-2019),
* if one operand is a QNaN, and the other
* operand is numerical, then return numerical argument.
*/
if ((flags & (minmax_isnum | minmax_isnumber))
&& !(ab_mask & float_cmask_snan)
&& (ab_mask & ~float_cmask_qnan)) {
return is_nan(a->cls) ? b : a;
}
/*
* In IEEE 754-2019, minNum, maxNum, minNumMag and maxNumMag
* are removed and replaced with minimum, minimumNumber, maximum
* and maximumNumber.
* minimumNumber/maximumNumber behavior for SNaN is changed to:
* If both operands are NaNs, a QNaN is returned.
* If either operand is a SNaN,
* an invalid operation exception is signaled,
* but unless both operands are NaNs,
* the SNaN is otherwise ignored and not converted to a QNaN.
*/
if ((flags & minmax_isnumber)
&& (ab_mask & float_cmask_snan)
&& (ab_mask & ~float_cmask_anynan)) {
float_raise(float_flag_invalid, s);
return is_nan(a->cls) ? b : a;
}
return parts_pick_nan(a, b, s);
}
a_exp = a->exp;
b_exp = b->exp;
if (unlikely(ab_mask != float_cmask_normal)) {
switch (a->cls) {
case float_class_normal:
break;
case float_class_inf:
a_exp = INT16_MAX;
break;
case float_class_zero:
a_exp = INT16_MIN;
break;
default:
g_assert_not_reached();
break;
}
switch (b->cls) {
case float_class_normal:
break;
case float_class_inf:
b_exp = INT16_MAX;
break;
case float_class_zero:
b_exp = INT16_MIN;
break;
default:
g_assert_not_reached();
break;
}
}
/* Compare magnitudes. */
cmp = a_exp - b_exp;
if (cmp == 0) {
cmp = frac_cmp(a, b);
}
/*
* Take the sign into account.
* For ismag, only do this if the magnitudes are equal.
*/
if (!(flags & minmax_ismag) || cmp == 0) {
if (a->sign != b->sign) {
/* For differing signs, the negative operand is less. */
cmp = a->sign ? -1 : 1;
} else if (a->sign) {
/* For two negative operands, invert the magnitude comparison. */
cmp = -cmp;
}
}
if (flags & minmax_ismin) {
cmp = -cmp;
}
return cmp < 0 ? b : a;
}
/*
* Floating point compare
*/
static FloatRelation partsN(compare)(FloatPartsN *a, FloatPartsN *b,
float_status *s, bool is_quiet)
{
int ab_mask = float_cmask(a->cls) | float_cmask(b->cls);
if (likely(ab_mask == float_cmask_normal)) {
FloatRelation cmp;
if (a->sign != b->sign) {
goto a_sign;
}
if (a->exp == b->exp) {
cmp = frac_cmp(a, b);
} else if (a->exp < b->exp) {
cmp = float_relation_less;
} else {
cmp = float_relation_greater;
}
if (a->sign) {
cmp = -cmp;
}
return cmp;
}
if (unlikely(ab_mask & float_cmask_anynan)) {
if (ab_mask & float_cmask_snan) {
float_raise(float_flag_invalid | float_flag_invalid_snan, s);
} else if (!is_quiet) {
float_raise(float_flag_invalid, s);
}
return float_relation_unordered;
}
if (ab_mask & float_cmask_zero) {
if (ab_mask == float_cmask_zero) {
return float_relation_equal;
} else if (a->cls == float_class_zero) {
goto b_sign;
} else {
goto a_sign;
}
}
if (ab_mask == float_cmask_inf) {
if (a->sign == b->sign) {
return float_relation_equal;
}
} else if (b->cls == float_class_inf) {
goto b_sign;
} else {
g_assert(a->cls == float_class_inf);
}
a_sign:
return a->sign ? float_relation_less : float_relation_greater;
b_sign:
return b->sign ? float_relation_greater : float_relation_less;
}
/*
* Multiply A by 2 raised to the power N.
*/
static void partsN(scalbn)(FloatPartsN *a, int n, float_status *s)
{
switch (a->cls) {
case float_class_snan:
case float_class_qnan:
parts_return_nan(a, s);
break;
case float_class_zero:
case float_class_inf:
break;
case float_class_normal:
a->exp += MIN(MAX(n, -0x10000), 0x10000);
break;
default:
g_assert_not_reached();
}
}
/*
* Return log2(A)
*/
static void partsN(log2)(FloatPartsN *a, float_status *s, const FloatFmt *fmt)
{
uint64_t a0, a1, r, t, ign;
FloatPartsN f;
int i, n, a_exp, f_exp;
if (unlikely(a->cls != float_class_normal)) {
switch (a->cls) {
case float_class_snan:
case float_class_qnan:
parts_return_nan(a, s);
return;
case float_class_zero:
float_raise(float_flag_divbyzero, s);
/* log2(0) = -inf */
a->cls = float_class_inf;
a->sign = 1;
return;
case float_class_inf:
if (unlikely(a->sign)) {
goto d_nan;
}
return;
default:
break;
}
g_assert_not_reached();
}
if (unlikely(a->sign)) {
goto d_nan;
}
/* TODO: This algorithm looses bits too quickly for float128. */
g_assert(N == 64);
a_exp = a->exp;
f_exp = -1;
r = 0;
t = DECOMPOSED_IMPLICIT_BIT;
a0 = a->frac_hi;
a1 = 0;
n = fmt->frac_size + 2;
if (unlikely(a_exp == -1)) {
/*
* When a_exp == -1, we're computing the log2 of a value [0.5,1.0).
* When the value is very close to 1.0, there are lots of 1's in
* the msb parts of the fraction. At the end, when we subtract
* this value from -1.0, we can see a catastrophic loss of precision,
* as 0x800..000 - 0x7ff..ffx becomes 0x000..00y, leaving only the
* bits of y in the final result. To minimize this, compute as many
* digits as we can.
* ??? This case needs another algorithm to avoid this.
*/
n = fmt->frac_size * 2 + 2;
/* Don't compute a value overlapping the sticky bit */
n = MIN(n, 62);
}
for (i = 0; i < n; i++) {
if (a1) {
mul128To256(a0, a1, a0, a1, &a0, &a1, &ign, &ign);
} else if (a0 & 0xffffffffull) {
mul64To128(a0, a0, &a0, &a1);
} else if (a0 & ~DECOMPOSED_IMPLICIT_BIT) {
a0 >>= 32;
a0 *= a0;
} else {
goto exact;
}
if (a0 & DECOMPOSED_IMPLICIT_BIT) {
if (unlikely(a_exp == 0 && r == 0)) {
/*
* When a_exp == 0, we're computing the log2 of a value
* [1.0,2.0). When the value is very close to 1.0, there
* are lots of 0's in the msb parts of the fraction.
* We need to compute more digits to produce a correct
* result -- restart at the top of the fraction.
* ??? This is likely to lose precision quickly, as for
* float128; we may need another method.
*/
f_exp -= i;
t = r = DECOMPOSED_IMPLICIT_BIT;
i = 0;
} else {
r |= t;
}
} else {
add128(a0, a1, a0, a1, &a0, &a1);
}
t >>= 1;
}
/* Set sticky for inexact. */
r |= (a1 || a0 & ~DECOMPOSED_IMPLICIT_BIT);
exact:
parts_sint_to_float(a, a_exp, 0, s);
if (r == 0) {
return;
}
memset(&f, 0, sizeof(f));
f.cls = float_class_normal;
f.frac_hi = r;
f.exp = f_exp - frac_normalize(&f);
if (a_exp < 0) {
parts_sub_normal(a, &f);
} else if (a_exp > 0) {
parts_add_normal(a, &f);
} else {
*a = f;
}
return;
d_nan:
float_raise(float_flag_invalid, s);
parts_default_nan(a, s);
}
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