math.log/1 on complex, real part

Percentage Accurate: 52.0% → 100.0%

Time: 12.1s

Alternatives: 8

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return log(hypot(re, im));
}

Java

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

Python

def code(re, im):
	return math.log(math.hypot(re, im))

Julia

function code(re, im)
	return log(hypot(re, im))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im));
end

Wolfram

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

Derivation

1. Initial program 49.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. lower-hypot.f64 N/A

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

   8. unpow1 N/A

      \[\leadsto \log \left(\mathsf{hypot}\left(\color{blue}{{\left(\mathsf{neg}\left(re\right)\right)}^{1}}, \mathsf{neg}\left(im\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \log \left(\mathsf{hypot}\left({\left(\mathsf{neg}\left(re\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}, \mathsf{neg}\left(im\right)\right)\right) \]

   10. sqrt-pow1 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(\color{blue}{\sqrt{{\left(\mathsf{neg}\left(re\right)\right)}^{2}}}, \mathsf{neg}\left(im\right)\right)\right) \]

   11. pow2 N/A

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

   12. sqr-neg-rev N/A

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

   13. pow2 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(\sqrt{\color{blue}{{re}^{2}}}, \mathsf{neg}\left(im\right)\right)\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(\color{blue}{{re}^{\left(\frac{2}{2}\right)}}, \mathsf{neg}\left(im\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \log \left(\mathsf{hypot}\left({re}^{\color{blue}{1}}, \mathsf{neg}\left(im\right)\right)\right) \]

   16. unpow1 N/A

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

   17. unpow1 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(re, \color{blue}{{\left(\mathsf{neg}\left(im\right)\right)}^{1}}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(re, {\left(\mathsf{neg}\left(im\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)\right) \]

   19. sqrt-pow1 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(re, \color{blue}{\sqrt{{\left(\mathsf{neg}\left(im\right)\right)}^{2}}}\right)\right) \]

   20. pow2 N/A

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

   21. sqr-neg-rev N/A

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

   22. pow2 N/A

       \[\leadsto \log \left(\mathsf{hypot}\left(re, \sqrt{\color{blue}{{im}^{2}}}\right)\right) \]

   23. sqrt-pow1 N/A

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

   24. metadata-eval N/A

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

   25. unpow1 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 25.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

Alternative 3: 27.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return log(im);
}

Fortran

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

Java

public static double code(double re, double im) {
	return Math.log(im);
}

Python

def code(re, im):
	return math.log(im)

Julia

function code(re, im)
	return log(im)
end

MATLAB

function tmp = code(re, im)
	tmp = log(im);
end

Wolfram

code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

Alternative 4: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return (((0.5 / im) * re) * re) / im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 2.9%

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

13. Applied rewrites 3.2%

    \[\leadsto \frac{\left(\frac{0.5}{im} \cdot re\right) \cdot re}{im} \]
14. Add Preprocessing

Alternative 5: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return (re / im) * ((re / im) * 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 2.9%

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

13. Applied rewrites 3.2%

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

Alternative 6: 3.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return ((re / im) * (0.5 / im)) * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 2.9%

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

13. Applied rewrites 3.2%

    \[\leadsto \left(\frac{re}{im} \cdot \frac{0.5}{im}\right) \cdot re \]
14. Add Preprocessing

Alternative 7: 3.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot re}{im \cdot im} \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return ((0.5 * re) / (im * im)) * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 2.9%

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

13. Applied rewrites 2.9%

    \[\leadsto \frac{0.5 \cdot re}{im \cdot im} \cdot re \]
14. Add Preprocessing

Alternative 8: 3.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return ((0.5 / (im * im)) * re) * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 25.5

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 2.9%

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

Reproduce

herbie shell --seed 2024319 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))