math.abs on complex

Percentage Accurate: 53.7% → 100.0%

Time: 5.8s

Alternatives: 4

Speedup: 1.0×

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: 53.7% 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 57.0%

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

   1. hypot-define N/A

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

   2. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{hypot.f64}\left(re, \color{blue}{im}\right) \]

3. Simplified 100.0%

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

Alternative 2: 26.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ im + \frac{re}{\frac{im}{re}} \cdot \left(0.5 + \frac{\frac{re}{im}}{\frac{im}{\frac{re}{-8}}}\right) \end{array} \]

FPCore

(FPCore modulus (re im)
 :precision binary64
 (+ im (* (/ re (/ im re)) (+ 0.5 (/ (/ re im) (/ im (/ re -8.0)))))))

C

double modulus(double re, double im) {
	return im + ((re / (im / re)) * (0.5 + ((re / im) / (im / (re / -8.0)))));
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = im + ((re / (im / re)) * (0.5d0 + ((re / im) / (im / (re / (-8.0d0))))))
end function

Java

public static double modulus(double re, double im) {
	return im + ((re / (im / re)) * (0.5 + ((re / im) / (im / (re / -8.0)))));
}

Python

def modulus(re, im):
	return im + ((re / (im / re)) * (0.5 + ((re / im) / (im / (re / -8.0)))))

Julia

function modulus(re, im)
	return Float64(im + Float64(Float64(re / Float64(im / re)) * Float64(0.5 + Float64(Float64(re / im) / Float64(im / Float64(re / -8.0))))))
end

MATLAB

function tmp = modulus(re, im)
	tmp = im + ((re / (im / re)) * (0.5 + ((re / im) / (im / (re / -8.0)))));
end

Wolfram

modulus[re_, im_] := N[(im + N[(N[(re / N[(im / re), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re / im), $MachinePrecision] / N[(im / N[(re / -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. hypot-define N/A

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

   2. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{hypot.f64}\left(re, \color{blue}{im}\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)} \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right)}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot {re}^{2}}{{im}^{3}}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot {re}^{2}\right), \left({im}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({re}^{2} \cdot \frac{-1}{8}\right), \left({im}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{8}\right), \left({im}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{8}\right), \left({im}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \left({im}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   12. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \left(im \cdot \left(im \cdot im\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \left(im \cdot {im}^{2}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \left({im}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \left(im \cdot im\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{im}}\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \left(\frac{\frac{1}{2}}{im}\right)\right)\right)\right) \]

   19. /-lowering-/.f64 19.8%

       \[\leadsto \mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right)\right)\right) \]

7. Simplified 19.8%

   \[\leadsto \color{blue}{im + \left(re \cdot re\right) \cdot \left(\frac{\left(re \cdot re\right) \cdot -0.125}{im \cdot \left(im \cdot im\right)} + \frac{0.5}{im}\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(re \cdot re\right) \cdot \left(\frac{\left(re \cdot re\right) \cdot \frac{-1}{8}}{im \cdot \left(im \cdot im\right)} + \frac{\frac{1}{2}}{im}\right) + \color{blue}{im} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(re \cdot re\right) \cdot \left(\frac{\left(re \cdot re\right) \cdot \frac{-1}{8}}{im \cdot \left(im \cdot im\right)} + \frac{\frac{1}{2}}{im}\right)\right), \color{blue}{im}\right) \]

9. Applied egg-rr 21.5%

   \[\leadsto \color{blue}{\frac{re}{\frac{im}{re}} \cdot \left(0.5 + \frac{re \cdot re}{\frac{im \cdot im}{-0.125}}\right) + im} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\frac{\frac{im \cdot im}{\frac{-1}{8}}}{re \cdot re}}\right)\right)\right), im\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\frac{im \cdot \frac{im}{\frac{-1}{8}}}{re \cdot re}}\right)\right)\right), im\right) \]

    3. times-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\frac{im}{re} \cdot \frac{\frac{im}{\frac{-1}{8}}}{re}}\right)\right)\right), im\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{\frac{im}{re}}}{\frac{\frac{im}{\frac{-1}{8}}}{re}}\right)\right)\right), im\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{re}{im}}{\frac{\frac{im}{\frac{-1}{8}}}{re}}\right)\right)\right), im\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{re}{im}\right), \left(\frac{\frac{im}{\frac{-1}{8}}}{re}\right)\right)\right)\right), im\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{\frac{im}{\frac{-1}{8}}}{re}\right)\right)\right)\right), im\right) \]

    8. associate-/l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{im}{re \cdot \frac{-1}{8}}\right)\right)\right)\right), im\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{im}{re \cdot \frac{1}{-8}}\right)\right)\right)\right), im\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{im}{re \cdot \frac{1}{\frac{1}{\frac{-1}{8}}}}\right)\right)\right)\right), im\right) \]

    11. div-inv N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{im}{\frac{re}{\frac{1}{\frac{-1}{8}}}}\right)\right)\right)\right), im\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \mathsf{/.f64}\left(im, \left(\frac{re}{\frac{1}{\frac{-1}{8}}}\right)\right)\right)\right)\right), im\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \mathsf{/.f64}\left(im, \mathsf{/.f64}\left(re, \left(\frac{1}{\frac{-1}{8}}\right)\right)\right)\right)\right)\right), im\right) \]

    14. metadata-eval 24.1%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), \mathsf{/.f64}\left(im, \mathsf{/.f64}\left(re, -8\right)\right)\right)\right)\right), im\right) \]

11. Applied egg-rr 24.1%

    \[\leadsto \frac{re}{\frac{im}{re}} \cdot \left(0.5 + \color{blue}{\frac{\frac{re}{im}}{\frac{im}{\frac{re}{-8}}}}\right) + im \]

12. Final simplification 24.1%

    \[\leadsto im + \frac{re}{\frac{im}{re}} \cdot \left(0.5 + \frac{\frac{re}{im}}{\frac{im}{\frac{re}{-8}}}\right) \]
13. Add Preprocessing

Alternative 3: 27.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = im + (0.5d0 * (re * (re / im)))
end function

Java

public static double modulus(double re, double im) {
	return im + (0.5 * (re * (re / im)));
}

Python

def modulus(re, im):
	return im + (0.5 * (re * (re / im)))

Julia

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

MATLAB

function tmp = modulus(re, im)
	tmp = im + (0.5 * (re * (re / im)));
end

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. hypot-define N/A

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

   2. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{hypot.f64}\left(re, \color{blue}{im}\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. *-inverses N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*l/ N/A

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

   10. times-frac N/A

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

   11. unpow2 N/A

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

   12. associate-*l/ N/A

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

   13. associate-*r/ N/A

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

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right) \cdot im\right)}\right) \]

   15. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}} \cdot im\right)\right) \]

   16. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{\color{blue}{{im}^{2}}}\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{im \cdot \color{blue}{im}}\right)\right) \]

   18. times-frac N/A

       \[\leadsto \mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot {re}^{2}}{im} \cdot \color{blue}{\frac{im}{im}}\right)\right) \]

7. Simplified 24.6%

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

Alternative 4: 26.9% 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 57.0%

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

   1. hypot-define N/A

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

   2. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{hypot.f64}\left(re, \color{blue}{im}\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
6. Step-by-step derivation

7. Simplified 24.2%

   \[\leadsto \color{blue}{im} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))