math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%

Time: 1.7s

Alternatives: 4

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 44.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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-udef 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 2: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{-131} \lor \neg \left(im \leq 4.75 \cdot 10^{-85}\right) \land im \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore modulus_sqr (re im)
 :precision binary64
 (if (or (<= im 7.5e-131) (and (not (<= im 4.75e-85)) (<= im 4.3e-29)))
   (* re re)
   (* im im)))

C

double modulus_sqr(double re, double im) {
	double tmp;
	if ((im <= 7.5e-131) || (!(im <= 4.75e-85) && (im <= 4.3e-29))) {
		tmp = re * re;
	} else {
		tmp = im * im;
	}
	return tmp;
}

Fortran

real(8) function modulus_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 7.5d-131) .or. (.not. (im <= 4.75d-85)) .and. (im <= 4.3d-29)) then
        tmp = re * re
    else
        tmp = im * im
    end if
    modulus_sqr = tmp
end function

Java

public static double modulus_sqr(double re, double im) {
	double tmp;
	if ((im <= 7.5e-131) || (!(im <= 4.75e-85) && (im <= 4.3e-29))) {
		tmp = re * re;
	} else {
		tmp = im * im;
	}
	return tmp;
}

Python

def modulus_sqr(re, im):
	tmp = 0
	if (im <= 7.5e-131) or (not (im <= 4.75e-85) and (im <= 4.3e-29)):
		tmp = re * re
	else:
		tmp = im * im
	return tmp

Julia

function modulus_sqr(re, im)
	tmp = 0.0
	if ((im <= 7.5e-131) || (!(im <= 4.75e-85) && (im <= 4.3e-29)))
		tmp = Float64(re * re);
	else
		tmp = Float64(im * im);
	end
	return tmp
end

MATLAB

function tmp_2 = modulus_sqr(re, im)
	tmp = 0.0;
	if ((im <= 7.5e-131) || (~((im <= 4.75e-85)) && (im <= 4.3e-29)))
		tmp = re * re;
	else
		tmp = im * im;
	end
	tmp_2 = tmp;
end

Wolfram

modulus$95$sqr[re_, im_] := If[Or[LessEqual[im, 7.5e-131], And[N[Not[LessEqual[im, 4.75e-85]], $MachinePrecision], LessEqual[im, 4.3e-29]]], N[(re * re), $MachinePrecision], N[(im * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.49999999999999964e-131 or 4.74999999999999982e-85 < im < 4.2999999999999998e-29

   1. Initial program 100.0%

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

   2. Taylor expanded in re around inf 66.7%

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

      1. unpow2 66.7%

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

   4. Simplified 66.7%

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

   if 7.49999999999999964e-131 < im < 4.74999999999999982e-85 or 4.2999999999999998e-29 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 71.7%

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

      1. unpow2 71.7%

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

   4. Simplified 71.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{-131} \lor \neg \left(im \leq 4.75 \cdot 10^{-85}\right) \land im \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]

Alternative 3: 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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 58.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 53.1%

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

   1. unpow2 53.1%

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

4. Simplified 53.1%

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

5. Final simplification 53.1%

   \[\leadsto im \cdot im \]

Reproduce

herbie shell --seed 2023171 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))