math.abs on complex

Percentage Accurate: 54.1% → 100.0%

Time: 2.1s

Alternatives: 4

Speedup: 1.0×

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

Specification

Math

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

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

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

Fortran

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

Java

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

Python

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

Julia

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

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

Fortran

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

Java

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

Python

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

Julia

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore modulus (re im) :precision binary64 (hypot re im))

C

double modulus(double re, double im) {
	return hypot(re, im);
}

Java

public static double modulus(double re, double im) {
	return Math.hypot(re, im);
}

Python

def modulus(re, im):
	return math.hypot(re, im)

Julia

function modulus(re, im)
	return hypot(re, im)
end

MATLAB

function tmp = modulus(re, im)
	tmp = hypot(re, im);
end

Wolfram

modulus[re_, im_] := N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]

Derivation

1. Initial program 53.2%

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

   1. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{re \cdot re + im \cdot im}\right)\right)} \]

   2. expm1-udef 32.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{re \cdot re + im \cdot im}\right)} - 1} \]

   3. log1p-udef 32.2%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{re \cdot re + im \cdot im}\right)}} - 1 \]

   4. add-exp-log 34.3%

      \[\leadsto \color{blue}{\left(1 + \sqrt{re \cdot re + im \cdot im}\right)} - 1 \]

   5. hypot-def 75.2%

      \[\leadsto \left(1 + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right) - 1 \]

3. Applied egg-rr 75.2%

   \[\leadsto \color{blue}{\left(1 + \mathsf{hypot}\left(re, im\right)\right) - 1} \]
4. Step-by-step derivation

   1. +-commutative 75.2%

      \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(re, im\right) + 1\right)} - 1 \]

   2. associate--l+ 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(re, im\right) + \left(1 - 1\right)} \]

   3. metadata-eval 100.0%

      \[\leadsto \mathsf{hypot}\left(re, im\right) + \color{blue}{0} \]

   4. +-rgt-identity 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 44.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{im}{\frac{re}{im}} \cdot -0.5 - re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore modulus (re im)
 :precision binary64
 (if (<= im 2.5e-165) (- (* (/ im (/ re im)) -0.5) re) im))

C

double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = ((im / (re / im)) * -0.5) - re;
	} else {
		tmp = im;
	}
	return tmp;
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = ((im / (re / im)) * (-0.5d0)) - re
    else
        tmp = im
    end if
    modulus = tmp
end function

Java

public static double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = ((im / (re / im)) * -0.5) - re;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def modulus(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = ((im / (re / im)) * -0.5) - re
	else:
		tmp = im
	return tmp

Julia

function modulus(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(Float64(Float64(im / Float64(re / im)) * -0.5) - re);
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = modulus(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = ((im / (re / im)) * -0.5) - re;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

modulus[re_, im_] := If[LessEqual[im, 2.5e-165], N[(N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - re), $MachinePrecision], im]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 51.9%

      \[\sqrt{re \cdot re + im \cdot im} \]

   2. Taylor expanded in re around -inf 27.9%

      \[\leadsto \color{blue}{-1 \cdot re + -0.5 \cdot \frac{{im}^{2}}{re}} \]
   3. Step-by-step derivation

      1. +-commutative 27.9%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{im}^{2}}{re} + -1 \cdot re} \]

      2. mul-1-neg 27.9%

         \[\leadsto -0.5 \cdot \frac{{im}^{2}}{re} + \color{blue}{\left(-re\right)} \]

      3. unsub-neg 27.9%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{im}^{2}}{re} - re} \]

      4. *-commutative 27.9%

         \[\leadsto \color{blue}{\frac{{im}^{2}}{re} \cdot -0.5} - re \]

      5. unpow2 27.9%

         \[\leadsto \frac{\color{blue}{im \cdot im}}{re} \cdot -0.5 - re \]

      6. associate-/l* 29.6%

         \[\leadsto \color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5 - re \]

   4. Simplified 29.6%

      \[\leadsto \color{blue}{\frac{im}{\frac{re}{im}} \cdot -0.5 - re} \]

   if 2.4999999999999999e-165 < im

   1. Initial program 55.5%

      \[\sqrt{re \cdot re + im \cdot im} \]

   2. Taylor expanded in re around 0 61.3%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{im}{\frac{re}{im}} \cdot -0.5 - re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 3: 43.8% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (if (<= im 2.5e-165) (- re) im))

C

double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = -re;
	} else {
		tmp = im;
	}
	return tmp;
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = -re
    else
        tmp = im
    end if
    modulus = tmp
end function

Java

public static double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = -re;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def modulus(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = -re
	else:
		tmp = im
	return tmp

Julia

function modulus(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(-re);
	else
		tmp = im;
	end
	return tmp
end

MATLAB

function tmp_2 = modulus(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = -re;
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

modulus[re_, im_] := If[LessEqual[im, 2.5e-165], (-re), im]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 51.9%

      \[\sqrt{re \cdot re + im \cdot im} \]

   2. Taylor expanded in re around -inf 29.2%

      \[\leadsto \color{blue}{-1 \cdot re} \]
   3. Step-by-step derivation

      1. mul-1-neg 29.2%

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

   4. Simplified 29.2%

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

   if 2.4999999999999999e-165 < im

   1. Initial program 55.5%

      \[\sqrt{re \cdot re + im \cdot im} \]

   2. Taylor expanded in re around 0 61.3%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 4: 27.7% accurate, 107.0× speedup

Math

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

FPCore

(FPCore modulus (re im) :precision binary64 im)

C

double modulus(double re, double im) {
	return im;
}

Fortran

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

Java

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

Python

def modulus(re, im):
	return im

Julia

function modulus(re, im)
	return im
end

MATLAB

function tmp = modulus(re, im)
	tmp = im;
end

Wolfram

modulus[re_, im_] := im

Derivation

1. Initial program 53.2%

   \[\sqrt{re \cdot re + im \cdot im} \]

2. Taylor expanded in re around 0 23.2%

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

3. Final simplification 23.2%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023187 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))