math.abs on complex

Percentage Accurate: 54.6% → 100.0%

Time: 5.5s

Alternatives: 6

Speedup: 0.2×

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: 54.6% 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, 0.2× 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 51.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 48.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 0 \lor \neg \left(im \cdot im \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{im}{re} \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore modulus (re im)
 :precision binary64
 (if (or (<= (* im im) 0.0) (not (<= (* im im) 5e+300)))
   (* (/ im re) re)
   (sqrt (fma re re (* im im)))))

C

double modulus(double re, double im) {
	double tmp;
	if (((im * im) <= 0.0) || !((im * im) <= 5e+300)) {
		tmp = (im / re) * re;
	} else {
		tmp = sqrt(fma(re, re, (im * im)));
	}
	return tmp;
}

Julia

function modulus(re, im)
	tmp = 0.0
	if ((Float64(im * im) <= 0.0) || !(Float64(im * im) <= 5e+300))
		tmp = Float64(Float64(im / re) * re);
	else
		tmp = sqrt(fma(re, re, Float64(im * im)));
	end
	return tmp
end

Wolfram

modulus[re_, im_] := If[Or[LessEqual[N[(im * im), $MachinePrecision], 0.0], N[Not[LessEqual[N[(im * im), $MachinePrecision], 5e+300]], $MachinePrecision]], N[(N[(im / re), $MachinePrecision] * re), $MachinePrecision], N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 0.0 or 5.00000000000000026e300 < (*.f64 im im)

   1. Initial program 25.4%

      \[\sqrt{re \cdot re + im \cdot im} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{1}{im}, {re}^{2}, im\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{im}}, {re}^{2}, im\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{im}, {re}^{2}, im\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{im}}, {re}^{2}, im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{im}, \color{blue}{re \cdot re}, im\right) \]

      10. lower-*.f64 25.5

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{im}, \color{blue}{re \cdot re}, im\right) \]

   5. Applied rewrites 25.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)} \]

   6. Taylor expanded in re around inf

      \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im} + \frac{im}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto \frac{im}{re} \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.8%

       \[\leadsto \frac{im}{re} \cdot re \]

   if 0.0 < (*.f64 im im) < 5.00000000000000026e300

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 76.6

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

   4. Applied rewrites 76.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 0 \lor \neg \left(im \cdot im \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{im}{re} \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 0 \lor \neg \left(im \cdot im \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{im}{re} \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore modulus (re im)
 :precision binary64
 (if (or (<= (* im im) 0.0) (not (<= (* im im) 5e+300)))
   (* (/ im re) re)
   (sqrt (* im im))))

C

double modulus(double re, double im) {
	double tmp;
	if (((im * im) <= 0.0) || !((im * im) <= 5e+300)) {
		tmp = (im / re) * re;
	} else {
		tmp = sqrt((im * 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 * im) <= 0.0d0) .or. (.not. ((im * im) <= 5d+300))) then
        tmp = (im / re) * re
    else
        tmp = sqrt((im * im))
    end if
    modulus = tmp
end function

Java

public static double modulus(double re, double im) {
	double tmp;
	if (((im * im) <= 0.0) || !((im * im) <= 5e+300)) {
		tmp = (im / re) * re;
	} else {
		tmp = Math.sqrt((im * im));
	}
	return tmp;
}

Python

def modulus(re, im):
	tmp = 0
	if ((im * im) <= 0.0) or not ((im * im) <= 5e+300):
		tmp = (im / re) * re
	else:
		tmp = math.sqrt((im * im))
	return tmp

Julia

function modulus(re, im)
	tmp = 0.0
	if ((Float64(im * im) <= 0.0) || !(Float64(im * im) <= 5e+300))
		tmp = Float64(Float64(im / re) * re);
	else
		tmp = sqrt(Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = modulus(re, im)
	tmp = 0.0;
	if (((im * im) <= 0.0) || ~(((im * im) <= 5e+300)))
		tmp = (im / re) * re;
	else
		tmp = sqrt((im * im));
	end
	tmp_2 = tmp;
end

Wolfram

modulus[re_, im_] := If[Or[LessEqual[N[(im * im), $MachinePrecision], 0.0], N[Not[LessEqual[N[(im * im), $MachinePrecision], 5e+300]], $MachinePrecision]], N[(N[(im / re), $MachinePrecision] * re), $MachinePrecision], N[Sqrt[N[(im * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 0.0 or 5.00000000000000026e300 < (*.f64 im im)

   1. Initial program 25.4%

      \[\sqrt{re \cdot re + im \cdot im} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{1}{im}, {re}^{2}, im\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{im}}, {re}^{2}, im\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{im}, {re}^{2}, im\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{im}}, {re}^{2}, im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{im}, \color{blue}{re \cdot re}, im\right) \]

      10. lower-*.f64 25.5

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{im}, \color{blue}{re \cdot re}, im\right) \]

   5. Applied rewrites 25.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)} \]

   6. Taylor expanded in re around inf

      \[\leadsto {re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im} + \frac{im}{{re}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto \frac{im}{re} \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.8%

       \[\leadsto \frac{im}{re} \cdot re \]

   if 0.0 < (*.f64 im im) < 5.00000000000000026e300

   1. Initial program 76.6%

      \[\sqrt{re \cdot re + im \cdot im} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 0 \lor \neg \left(im \cdot im \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{im}{re} \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{im} \cdot re, re, im\right) \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (fma (* (/ 0.5 im) re) re im))

C

double modulus(double re, double im) {
	return fma(((0.5 / im) * re), re, im);
}

Julia

function modulus(re, im)
	return fma(Float64(Float64(0.5 / im) * re), re, im)
end

Wolfram

modulus[re_, im_] := N[(N[(N[(0.5 / im), $MachinePrecision] * re), $MachinePrecision] * re + im), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{1}{im}, {re}^{2}, im\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{im}}, {re}^{2}, im\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{im}, {re}^{2}, im\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{im}}, {re}^{2}, im\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{im}, \color{blue}{re \cdot re}, im\right) \]

   10. lower-*.f64 27.0

       \[\leadsto \mathsf{fma}\left(\frac{0.5}{im}, \color{blue}{re \cdot re}, im\right) \]

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)} \]
6. Step-by-step derivation

7. Applied rewrites 28.1%

   \[\leadsto \mathsf{fma}\left(\frac{0.5}{im} \cdot re, \color{blue}{re}, im\right) \]
8. Add Preprocessing

Alternative 5: 29.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 32.2

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

5. Applied rewrites 32.2%

   \[\leadsto \sqrt{\color{blue}{im \cdot im}} \]
6. Add Preprocessing

Alternative 6: 26.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def modulus(re, im):
	return -re

Julia

function modulus(re, im)
	return Float64(-re)
end

MATLAB

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

Wolfram

modulus[re_, im_] := (-re)

Derivation

1. Initial program 51.6%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(re\right)} \]

   2. lower-neg.f64 28.4

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

5. Applied rewrites 28.4%

   \[\leadsto \color{blue}{-re} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))