math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%

Time: 1.6s

Alternatives: 5

Speedup: 1.0×

• 42.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]

2. Taylor expanded in re around 0 100.0%

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

   1. unpow2 100.0%

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

   2. unpow2 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-def 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [re, im] = \mathsf{sort}([re, im])\\ \\ im \cdot im + re \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

[re, im] = sort([re, im])
def modulus_sqr(re, im):
	return (im * im) + (re * re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]

2. Final simplification 100.0%

   \[\leadsto im \cdot im + re \cdot re \]

Alternative 4: 88.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [re, im] = \mathsf{sort}([re, im])\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \end{array} \]

FPCore

NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore modulus_sqr (re im)
 :precision binary64
 (if (<= im 1.25e-100) (* re re) (* im im)))

C

assert(re < im);
double modulus_sqr(double re, double im) {
	double tmp;
	if (im <= 1.25e-100) {
		tmp = re * re;
	} else {
		tmp = im * im;
	}
	return tmp;
}

Fortran

NOTE: re and im should be sorted in increasing order before calling this function.
real(8) function modulus_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.25d-100) then
        tmp = re * re
    else
        tmp = im * im
    end if
    modulus_sqr = tmp
end function

Java

assert re < im;
public static double modulus_sqr(double re, double im) {
	double tmp;
	if (im <= 1.25e-100) {
		tmp = re * re;
	} else {
		tmp = im * im;
	}
	return tmp;
}

Python

[re, im] = sort([re, im])
def modulus_sqr(re, im):
	tmp = 0
	if im <= 1.25e-100:
		tmp = re * re
	else:
		tmp = im * im
	return tmp

Julia

re, im = sort([re, im])
function modulus_sqr(re, im)
	tmp = 0.0
	if (im <= 1.25e-100)
		tmp = Float64(re * re);
	else
		tmp = Float64(im * im);
	end
	return tmp
end

MATLAB

re, im = num2cell(sort([re, im])){:}
function tmp_2 = modulus_sqr(re, im)
	tmp = 0.0;
	if (im <= 1.25e-100)
		tmp = re * re;
	else
		tmp = im * im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: re and im should be sorted in increasing order before calling this function.
modulus$95$sqr[re_, im_] := If[LessEqual[im, 1.25e-100], N[(re * re), $MachinePrecision], N[(im * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.25e-100

   1. Initial program 100.0%

      \[re \cdot re + im \cdot im \]

   2. Taylor expanded in re around inf 67.3%

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

      1. unpow2 67.3%

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

   4. Simplified 67.3%

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

   if 1.25e-100 < im

   1. Initial program 100.0%

      \[re \cdot re + im \cdot im \]

   2. Taylor expanded in re around 0 75.1%

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

      1. unpow2 75.1%

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

   4. Simplified 75.1%

      \[\leadsto \color{blue}{im \cdot im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]

Alternative 5: 56.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [re, im] = \mathsf{sort}([re, im])\\ \\ im \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

[re, im] = sort([re, im])
def modulus_sqr(re, im):
	return im * im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]

2. Taylor expanded in re around 0 55.3%

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

   1. unpow2 55.3%

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

4. Simplified 55.3%

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

5. Final simplification 55.3%

   \[\leadsto im \cdot im \]

Reproduce

herbie shell --seed 2023189 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))