float
__fmaf (float x, float y, float z)
{
fenv_t env;
/* Multiplication is always exact. */
double temp = (double) x * (double) y;
/* Ensure correct sign of an exact zero result by performing the
addition in the original rounding mode in that case. */
if (temp == -z)
return (float) temp + z;
union ieee754_double u;
libc_feholdexcept_setround (&env, FE_TOWARDZERO);
/* Perform addition with round to odd. */
u.d = temp + (double) z;
/* Ensure the addition is not scheduled after fetestexcept call. */
math_force_eval (u.d);
/* Reset rounding mode and test for inexact simultaneously. */
int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
u.ieee.mantissa1 |= j;
/* And finally truncation with round to nearest. */
return (float) u.d;
}
double
__fma (double x, double y, double z)
{
union ieee754_double u, v, w;
int adjust = 0;
u.d = x;
v.d = y;
w.d = z;
if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
{
/* If z is Inf, but x and y are finite, the result should be
z rather than NaN. */
if (w.ieee.exponent == 0x7ff
&& u.ieee.exponent != 0x7ff
&& v.ieee.exponent != 0x7ff)
return (z + x) + y;
/* If z is zero and x are y are nonzero, compute the result
as x * y to avoid the wrong sign of a zero result if x * y
underflows to 0. */
if (z == 0 && x != 0 && y != 0)
return x * y;
/* If x or y or z is Inf/NaN, or if x * y is zero, compute as
x * y + z. */
if (u.ieee.exponent == 0x7ff
|| v.ieee.exponent == 0x7ff
|| w.ieee.exponent == 0x7ff
|| x == 0
|| y == 0)
return x * y + z;
/* If fma will certainly overflow, compute as x * y. */
if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
return x * y;
/* If x * y is less than 1/4 of DBL_DENORM_MIN, neither the
result nor whether there is underflow depends on its exact
value, only on its sign. */
if (u.ieee.exponent + v.ieee.exponent
< IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
{
int neg = u.ieee.negative ^ v.ieee.negative;
double tiny = neg ? -0x1p-1074 : 0x1p-1074;
if (w.ieee.exponent >= 3)
return tiny + z;
/* Scaling up, adding TINY and scaling down produces the
correct result, because in round-to-nearest mode adding
TINY has no effect and in other modes double rounding is
harmless. But it may not produce required underflow
exceptions. */
v.d = z * 0x1p54 + tiny;
if (TININESS_AFTER_ROUNDING
? v.ieee.exponent < 55
: (w.ieee.exponent == 0
|| (w.ieee.exponent == 1
&& w.ieee.negative != neg
&& w.ieee.mantissa1 == 0
&& w.ieee.mantissa0 == 0)))
{
volatile double force_underflow = x * y;
(void) force_underflow;
}
return v.d * 0x1p-54;
}
if (u.ieee.exponent + v.ieee.exponent
>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
{
/* Compute 1p-53 times smaller result and multiply
at the end. */
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent -= DBL_MANT_DIG;
else
v.ieee.exponent -= DBL_MANT_DIG;
/* If x + y exponent is very large and z exponent is very small,
it doesn't matter if we don't adjust it. */
if (w.ieee.exponent > DBL_MANT_DIG)
w.ieee.exponent -= DBL_MANT_DIG;
adjust = 1;
}
else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
/* Similarly.
If z exponent is very large and x and y exponents are
very small, adjust them up to avoid spurious underflows,
rather than down. */
if (u.ieee.exponent + v.ieee.exponent
<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG)
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
}
else if (u.ieee.exponent > v.ieee.exponent)
{
if (u.ieee.exponent > DBL_MANT_DIG)
u.ieee.exponent -= DBL_MANT_DIG;
}
else if (v.ieee.exponent > DBL_MANT_DIG)
v.ieee.exponent -= DBL_MANT_DIG;
w.ieee.exponent -= DBL_MANT_DIG;
adjust = 1;
}
else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
u.ieee.exponent -= DBL_MANT_DIG;
if (v.ieee.exponent)
v.ieee.exponent += DBL_MANT_DIG;
else
v.d *= 0x1p53;
}
else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
v.ieee.exponent -= DBL_MANT_DIG;
if (u.ieee.exponent)
u.ieee.exponent += DBL_MANT_DIG;
else
u.d *= 0x1p53;
}
else /* if (u.ieee.exponent + v.ieee.exponent
<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
{
if (w.ieee.exponent)
w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
else
w.d *= 0x1p108;
adjust = -1;
}
/* Otherwise x * y should just affect inexact
and nothing else. */
}
x = u.d;
y = v.d;
z = w.d;
}
/* Ensure correct sign of exact 0 + 0. */
if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
return x * y + z;
fenv_t env;
libc_feholdexcept_setround (&env, FE_TONEAREST);
/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
double x1 = x * C;
double y1 = y * C;
double m1 = x * y;
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
double x2 = x - x1;
double y2 = y - y1;
double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
double a1 = z + m1;
double t1 = a1 - z;
double t2 = a1 - t1;
t1 = m1 - t1;
t2 = z - t2;
double a2 = t1 + t2;
/* Ensure the arithmetic is not scheduled after feclearexcept call. */
math_force_eval (m2);
math_force_eval (a2);
feclearexcept (FE_INEXACT);
/* If the result is an exact zero, ensure it has the correct sign. */
if (a1 == 0 && m2 == 0)
{
libc_feupdateenv (&env);
/* Ensure that round-to-nearest value of z + m1 is not reused. */
z = math_opt_barrier (z);
return z + m1;
}
libc_fesetround (FE_TOWARDZERO);
/* Perform m2 + a2 addition with round to odd. */
u.d = a2 + m2;
if (__glibc_unlikely (adjust < 0))
{
if ((u.ieee.mantissa1 & 1) == 0)
u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
v.d = a1 + u.d;
/* Ensure the addition is not scheduled after fetestexcept call. */
math_force_eval (v.d);
}
/* Reset rounding mode and test for inexact simultaneously. */
int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
if (__glibc_likely (adjust == 0))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
u.ieee.mantissa1 |= j;
/* Result is a1 + u.d. */
return a1 + u.d;
}
else if (__glibc_likely (adjust > 0))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
u.ieee.mantissa1 |= j;
/* Result is a1 + u.d, scaled up. */
return (a1 + u.d) * 0x1p53;
}
else
{
/* If a1 + u.d is exact, the only rounding happens during
scaling down. */
if (j == 0)
return v.d * 0x1p-108;
/* If result rounded to zero is not subnormal, no double
rounding will occur. */
if (v.ieee.exponent > 108)
return (a1 + u.d) * 0x1p-108;
/* If v.d * 0x1p-108 with round to zero is a subnormal above
or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
down just by 1 bit, which means v.ieee.mantissa1 |= j would
change the round bit, not sticky or guard bit.
v.d * 0x1p-108 never normalizes by shifting up,
so round bit plus sticky bit should be already enough
for proper rounding. */
if (v.ieee.exponent == 108)
{
/* If the exponent would be in the normal range when
rounding to normal precision with unbounded exponent
range, the exact result is known and spurious underflows
must be avoided on systems detecting tininess after
rounding. */
if (TININESS_AFTER_ROUNDING)
{
w.d = a1 + u.d;
if (w.ieee.exponent == 109)
return w.d * 0x1p-108;
}
/* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
v.ieee.mantissa1 & 1 is the round bit and j is our sticky
bit. */
w.d = 0.0;
w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
w.ieee.negative = v.ieee.negative;
v.ieee.mantissa1 &= ~3U;
v.d *= 0x1p-108;
w.d *= 0x1p-2;
return v.d + w.d;
}
v.ieee.mantissa1 |= j;
return v.d * 0x1p-108;
}
}
double
__fma (double x, double y, double z)
{
union ieee754_double u, v, w;
int adjust = 0;
u.d = x;
v.d = y;
w.d = z;
if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
{
/* If z is Inf, but x and y are finite, the result should be
z rather than NaN. */
if (w.ieee.exponent == 0x7ff
&& u.ieee.exponent != 0x7ff
&& v.ieee.exponent != 0x7ff)
return (z + x) + y;
/* If z is zero and x are y are nonzero, compute the result
as x * y to avoid the wrong sign of a zero result if x * y
underflows to 0. */
if (z == 0 && x != 0 && y != 0)
return x * y;
/* If x or y or z is Inf/NaN, or if fma will certainly overflow,
or if x * y is less than half of DBL_DENORM_MIN,
compute as x * y + z. */
if (u.ieee.exponent == 0x7ff
|| v.ieee.exponent == 0x7ff
|| w.ieee.exponent == 0x7ff
|| u.ieee.exponent + v.ieee.exponent
> 0x7ff + IEEE754_DOUBLE_BIAS
|| u.ieee.exponent + v.ieee.exponent
< IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
return x * y + z;
if (u.ieee.exponent + v.ieee.exponent
>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
{
/* Compute 1p-53 times smaller result and multiply
at the end. */
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent -= DBL_MANT_DIG;
else
v.ieee.exponent -= DBL_MANT_DIG;
/* If x + y exponent is very large and z exponent is very small,
it doesn't matter if we don't adjust it. */
if (w.ieee.exponent > DBL_MANT_DIG)
w.ieee.exponent -= DBL_MANT_DIG;
adjust = 1;
}
else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
/* Similarly.
If z exponent is very large and x and y exponents are
very small, it doesn't matter if we don't adjust it. */
if (u.ieee.exponent > v.ieee.exponent)
{
if (u.ieee.exponent > DBL_MANT_DIG)
u.ieee.exponent -= DBL_MANT_DIG;
}
else if (v.ieee.exponent > DBL_MANT_DIG)
v.ieee.exponent -= DBL_MANT_DIG;
w.ieee.exponent -= DBL_MANT_DIG;
adjust = 1;
}
else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
u.ieee.exponent -= DBL_MANT_DIG;
if (v.ieee.exponent)
v.ieee.exponent += DBL_MANT_DIG;
else
v.d *= 0x1p53;
}
else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
{
v.ieee.exponent -= DBL_MANT_DIG;
if (u.ieee.exponent)
u.ieee.exponent += DBL_MANT_DIG;
else
u.d *= 0x1p53;
}
else /* if (u.ieee.exponent + v.ieee.exponent
<= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * DBL_MANT_DIG;
else
v.ieee.exponent += 2 * DBL_MANT_DIG;
if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 4)
{
if (w.ieee.exponent)
w.ieee.exponent += 2 * DBL_MANT_DIG;
else
w.d *= 0x1p106;
adjust = -1;
}
/* Otherwise x * y should just affect inexact
and nothing else. */
}
x = u.d;
y = v.d;
z = w.d;
}
/* Ensure correct sign of exact 0 + 0. */
if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0))
return x * y + z;
/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
double x1 = x * C;
double y1 = y * C;
double m1 = x * y;
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
double x2 = x - x1;
double y2 = y - y1;
double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
double a1 = z + m1;
double t1 = a1 - z;
double t2 = a1 - t1;
t1 = m1 - t1;
t2 = z - t2;
double a2 = t1 + t2;
fenv_t env;
libc_feholdexcept_setround (&env, FE_TOWARDZERO);
/* Perform m2 + a2 addition with round to odd. */
u.d = a2 + m2;
if (__builtin_expect (adjust < 0, 0))
{
if ((u.ieee.mantissa1 & 1) == 0)
u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
v.d = a1 + u.d;
/* Ensure the addition is not scheduled after fetestexcept call. */
math_force_eval (v.d);
}
/* Reset rounding mode and test for inexact simultaneously. */
int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
if (__builtin_expect (adjust == 0, 1))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
u.ieee.mantissa1 |= j;
/* Result is a1 + u.d. */
return a1 + u.d;
}
else if (__builtin_expect (adjust > 0, 1))
{
if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
u.ieee.mantissa1 |= j;
/* Result is a1 + u.d, scaled up. */
return (a1 + u.d) * 0x1p53;
}
else
{
/* If a1 + u.d is exact, the only rounding happens during
scaling down. */
if (j == 0)
return v.d * 0x1p-106;
/* If result rounded to zero is not subnormal, no double
rounding will occur. */
if (v.ieee.exponent > 106)
return (a1 + u.d) * 0x1p-106;
/* If v.d * 0x1p-106 with round to zero is a subnormal above
or equal to DBL_MIN / 2, then v.d * 0x1p-106 shifts mantissa
down just by 1 bit, which means v.ieee.mantissa1 |= j would
change the round bit, not sticky or guard bit.
v.d * 0x1p-106 never normalizes by shifting up,
so round bit plus sticky bit should be already enough
for proper rounding. */
if (v.ieee.exponent == 106)
{
/* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
v.ieee.mantissa1 & 1 is the round bit and j is our sticky
bit. In round-to-nearest 001 rounds down like 00,
011 rounds up, even though 01 rounds down (thus we need
to adjust), 101 rounds down like 10 and 111 rounds up
like 11. */
if ((v.ieee.mantissa1 & 3) == 1)
{
v.d *= 0x1p-106;
if (v.ieee.negative)
return v.d - 0x1p-1074 /* __DBL_DENORM_MIN__ */;
else
return v.d + 0x1p-1074 /* __DBL_DENORM_MIN__ */;
}
else
return v.d * 0x1p-106;
}
v.ieee.mantissa1 |= j;
return v.d * 0x1p-106;
}
}
