math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 2

Speedup: 1.0×

• 44.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. pow2 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, {re}^{2}\right)} \]
5. Step-by-step derivation

   1. unpow2 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(im, im, \color{blue}{re \cdot re}\right) \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Reproduce

herbie shell --seed 2024149 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))