math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%

Time: 1.6s

Alternatives: 2

Speedup: 1.0×

• 43.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re + im \cdot im \end{array} \]

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

C

double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Fortran

real(8) function modulus_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus_sqr = (re * re) + (im * im)
end function

Java

public static double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Python

def modulus_sqr(re, im):
	return (re * re) + (im * im)

Julia

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end

MATLAB

function tmp = modulus_sqr(re, im)
	tmp = (re * re) + (im * im);
end

Wolfram

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re + im \cdot im \end{array} \]

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

C

double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Fortran

real(8) function modulus_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus_sqr = (re * re) + (im * im)
end function

Java

public static double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Python

def modulus_sqr(re, im):
	return (re * re) + (im * im)

Julia

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end

MATLAB

function tmp = modulus_sqr(re, im)
	tmp = (re * re) + (im * im);
end

Wolfram

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \mathsf{fma}\left(im_m, im_m, {re_m}^{2}\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus_sqr (re_m im_m)
 :precision binary64
 (fma im_m im_m (pow re_m 2.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double modulus_sqr(double re_m, double im_m) {
	return fma(im_m, im_m, pow(re_m, 2.0));
}

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function modulus_sqr(re_m, im_m)
	return fma(im_m, im_m, (re_m ^ 2.0))
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus$95$sqr[re$95$m_, im$95$m_] := N[(im$95$m * im$95$m + N[Power[re$95$m, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]
2. Add Preprocessing

3. Taylor expanded in re around 0 100.0%

   \[\leadsto \color{blue}{{im}^{2} + {re}^{2}} \]
4. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \color{blue}{im \cdot im} + {re}^{2} \]

   2. fma-udef 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, {re}^{2}\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, {re}^{2}\right)} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(im, im, {re}^{2}\right) \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ re_m \cdot re_m + im_m \cdot im_m \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus_sqr (re_m im_m)
 :precision binary64
 (+ (* re_m re_m) (* im_m im_m)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double modulus_sqr(double re_m, double im_m) {
	return (re_m * re_m) + (im_m * im_m);
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function modulus_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus_sqr = (re_m * re_m) + (im_m * im_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double modulus_sqr(double re_m, double im_m) {
	return (re_m * re_m) + (im_m * im_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def modulus_sqr(re_m, im_m):
	return (re_m * re_m) + (im_m * im_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function modulus_sqr(re_m, im_m)
	return Float64(Float64(re_m * re_m) + Float64(im_m * im_m))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = modulus_sqr(re_m, im_m)
	tmp = (re_m * re_m) + (im_m * im_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus$95$sqr[re$95$m_, im$95$m_] := N[(N[(re$95$m * re$95$m), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto re \cdot re + im \cdot im \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))