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222 | // These functions approximate reciprocal, square root, and
// cube root to varying degrees of precision substantially
// faster than the straightforward methods 1/x, sqrtf(x),
// and powf( x, 1.0f/3.0f ). All require SSE-enabled CPU & OS.
// Unlike the powf() solution, the cube roots also correctly
// handle negative inputs.
#define REALLY_INLINE __forceinline
// ~34 clocks on Pentium M vs. ~360 for powf
REALLY_INLINE float cuberoot_sse_8bits( float x )
{
float z;
static const float three = 3.0f;
_asm
{
mov eax, x // x as bits
movss xmm2, x // x2: x
movss xmm1, three // x1: 3
// Magic floating point cube root done with integer math.
// The exponent is divided by three in such a way that
// remainder bits get shoved into the top of the normalized
// mantissa.
mov ecx, eax // copy of x
and eax, 0x7FFFFFFF // exponent & mantissa of x in biased-127
sub eax, 0x3F800000 // exponent & mantissa of x in 2's comp
sar eax, 10 //
imul eax, 341 // 341/1024 ~= .333
add eax, 0x3F800000 // back to biased-127
and eax, 0x7FFFFFFF // remask
and ecx, 0x80000000 // original sign and mantissa
or eax, ecx // masked new exponent, old sign and mantissa
mov z, eax //
// use SSE to refine the first approximation
movss xmm0, z ;// x0: z
movss xmm3, xmm0 ;// x3: z
mulss xmm3, xmm0 ;// x3: z*z
movss xmm4, xmm3 ;// x4: z*z
mulss xmm3, xmm1 ;// x3: 3*z*z
rcpss xmm3, xmm3 ;// x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 ;// x4: z*z*z
subss xmm4, xmm2 ;// x4: z*z*z-x
mulss xmm4, xmm3 ;// x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 ;// x0: z' accurate to within about 0.3%
movss z, xmm0
}
return z;
}
// ~60 clocks on Pentium M vs. ~360 for powf
REALLY_INLINE float cuberoot_sse_16bits( float x )
{
float z;
static const float three = 3.0f;
_asm
{
mov eax, x // x as bits
movss xmm2, x // x2: x
movss xmm1, three // x1: 3
// Magic floating point cube root done with integer math.
// The exponent is divided by three in such a way that
// remainder bits get shoved into the top of the normalized
// mantissa.
mov ecx, eax // copy of x
and eax, 0x7FFFFFFF // exponent & mantissa of x in biased-127
sub eax, 0x3F800000 // exponent & mantissa of x in 2's comp
sar eax, 10 //
imul eax, 341 // 341/1024 ~= .333
add eax, 0x3F800000 // back to biased-127
and eax, 0x7FFFFFFF // remask
and ecx, 0x80000000 // original sign and mantissa
or eax, ecx // masked new exponent, old sign and mantissa
mov z, eax //
// use SSE to refine the first approximation
movss xmm0, z ;// x0: z
movss xmm3, xmm0 ;// x3: z
mulss xmm3, xmm0 ;// x3: z*z
movss xmm4, xmm3 ;// x4: z*z
mulss xmm3, xmm1 ;// x3: 3*z*z
rcpss xmm3, xmm3 ;// x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 ;// x4: z*z*z
subss xmm4, xmm2 ;// x4: z*z*z-x
mulss xmm4, xmm3 ;// x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 ;// x0: z' accurate to within about 0.3%
movss xmm3, xmm0 ;// x3: z
mulss xmm3, xmm0 ;// x3: z*z
movss xmm4, xmm3 ;// x4: z*z
mulss xmm3, xmm1 ;// x3: 3*z*z
rcpss xmm3, xmm3 ;// x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 ;// x4: z*z*z
subss xmm4, xmm2 ;// x4: z*z*z-x
mulss xmm4, xmm3 ;// x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 ;// x0: z'' accurate to within about 0.001%
movss z, xmm0
}
return z;
}
// ~77 clocks on Pentium M vs. ~360 for powf
REALLY_INLINE float cuberoot_sse_22bits( float x )
{
float z;
static const float three = 3.0f;
_asm
{
mov eax, x // x as bits
movss xmm2, x // x2: x
movss xmm1, three // x1: 3
// Magic floating point cube root done with integer math.
// The exponent is divided by three in such a way that
// remainder bits get shoved into the top of the normalized
// mantissa.
mov ecx, eax // copy of x
and eax, 0x7FFFFFFF // exponent & mantissa of x in biased-127
sub eax, 0x3F800000 // exponent & mantissa of x in 2's comp
sar eax, 10 //
imul eax, 341 // 341/1024 ~= .333
add eax, 0x3F800000 // back to biased-127
and eax, 0x7FFFFFFF // remask
and ecx, 0x80000000 // original sign and mantissa
or eax, ecx // masked new exponent, old sign and mantissa
mov z, eax //
// use SSE to refine the first approximation
movss xmm0, z // x0: z
movss xmm3, xmm0 // x3: z
mulss xmm3, xmm0 // x3: z*z
movss xmm4, xmm3 // x4: z*z
mulss xmm3, xmm1 // x3: 3*z*z
rcpss xmm3, xmm3 // x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 // x4: z*z*z
subss xmm4, xmm2 // x4: z*z*z-x
mulss xmm4, xmm3 // x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 // x0: z' accurate to within about 0.3%
movss xmm3, xmm0 // x3: z
mulss xmm3, xmm0 // x3: z*z
movss xmm4, xmm3 // x4: z*z
mulss xmm3, xmm1 // x3: 3*z*z
rcpss xmm3, xmm3 // x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 // x4: z*z*z
subss xmm4, xmm2 // x4: z*z*z-x
mulss xmm4, xmm3 // x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 // x0: z'' accurate to within about 0.001%
movss xmm3, xmm0 // x3: z
mulss xmm3, xmm0 // x3: z*z
movss xmm4, xmm3 // x4: z*z
mulss xmm3, xmm1 // x3: 3*z*z
rcpss xmm3, xmm3 // x3: ~ 1/(3*z*z)
mulss xmm4, xmm0 // x4: z*z*z
subss xmm4, xmm2 // x4: z*z*z-x
mulss xmm4, xmm3 // x4: (z*z*z-x) / (3*z*z)
subss xmm0, xmm4 // x0: z''' accurate to within about 0.000012%
movss z, xmm0
}
return z;
}
// ~6 clocks on Pentium M vs. ~24 for single precision sqrtf
REALLY_INLINE float squareroot_sse_11bits( float x )
{
float z;
_asm
{
rsqrtss xmm0, x
rcpss xmm0, xmm0
movss z, xmm0 // z ~= sqrt(x) to 0.038%
}
return z;
}
// ~19 clocks on Pentium M vs. ~24 for single precision sqrtf
REALLY_INLINE float squareroot_sse_22bits( float x )
{
static float half = 0.5f;
float z;
_asm
{
movss xmm1, x // x1: x
rsqrtss xmm2, xmm1 // x2: ~1/sqrt(x) = 1/z
rcpss xmm0, xmm2 // x0: z == ~sqrt(x) to 0.05%
movss xmm4, xmm0 // x4: z
movss xmm3, half
mulss xmm4, xmm4 // x4: z*z
mulss xmm2, xmm3 // x2: 1 / 2z
subss xmm4, xmm1 // x4: z*z-x
mulss xmm4, xmm2 // x4: (z*z-x)/2z
subss xmm0, xmm4 // x0: z' to 0.000015%
movss z, xmm0
}
return z;
}
// ~12 clocks on Pentium M vs. ~16 for single precision divide
REALLY_INLINE float reciprocal_sse_22bits( float x )
{
float z;
_asm
{
rcpss xmm0, x // x0: z ~= 1/x
movss xmm2, x // x2: x
movss xmm1, xmm0 // x1: z ~= 1/x
addss xmm0, xmm0 // x0: 2z
mulss xmm1, xmm1 // x1: z^2
mulss xmm1, xmm2 // x1: xz^2
subss xmm0, xmm1 // x0: z' ~= 1/x to 0.000012%
movss z, xmm0
}
return z;
}
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