math.log10 on complex, real part

Percentage Accurate: 51.7% → 99.0%

Time: 33.1s

Alternatives: 5

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (/ -1.0 (/ -1.0 (log (hypot im re)))) (log 10.0)))

C

double code(double re, double im) {
	return (-1.0 / (-1.0 / log(hypot(im, re)))) / log(10.0);
}

Java

public static double code(double re, double im) {
	return (-1.0 / (-1.0 / Math.log(Math.hypot(im, re)))) / Math.log(10.0);
}

Python

def code(re, im):
	return (-1.0 / (-1.0 / math.log(math.hypot(im, re)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(-1.0 / Float64(-1.0 / log(hypot(im, re)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (-1.0 / (-1.0 / log(hypot(im, re)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(-1.0 / N[(-1.0 / N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Step-by-step derivation

   1. lift-hypot.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   3. +-commutative N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}{\log 10} \]

   5. lift-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}{\log 10} \]

   6. remove-double-div N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

   7. unpow-1 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}}}{\log 10} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}}}{\log 10} \]

   9. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}}{\log 10} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}{\log 10} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}}{\log 10} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}\right)}}{\log 10} \]

   13. unpow-1 N/A

       \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}\right)}}{\log 10} \]

   14. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{\frac{\color{blue}{-1}}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}{\log 10} \]

   16. lower-/.f64 99.1

       \[\leadsto \frac{\frac{-1}{\color{blue}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

6. Applied rewrites 99.1%

   \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]
7. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Add Preprocessing

Alternative 3: 24.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \log im \cdot 2\right) \cdot \frac{-0.5}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (/ re im) (/ re im) (* (log im) 2.0)) (/ -0.5 (log 0.1))))

C

double code(double re, double im) {
	return fma((re / im), (re / im), (log(im) * 2.0)) * (-0.5 / log(0.1));
}

Julia

function code(re, im)
	return Float64(fma(Float64(re / im), Float64(re / im), Float64(log(im) * 2.0)) * Float64(-0.5 / log(0.1)))
end

Wolfram

code[re_, im_] := N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(N[Log[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)} \]

   4. lift-log.f64 N/A

      \[\leadsto \frac{1}{\log 10} \cdot \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\log 10} \cdot \log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)} \]

   6. pow1/2 N/A

      \[\leadsto \frac{1}{\log 10} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)} \]

   7. log-pow N/A

      \[\leadsto \frac{1}{\log 10} \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)\right)} \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\log 10} \cdot \frac{1}{2}\right) \cdot \log \left(re \cdot re + im \cdot im\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\log 10} \cdot \frac{1}{2}\right) \cdot \log \left(re \cdot re + im \cdot im\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{\log 10} \cdot \frac{1}{2}\right)} \cdot \log \left(re \cdot re + im \cdot im\right) \]

   11. inv-pow N/A

       \[\leadsto \left(\color{blue}{{\log 10}^{-1}} \cdot \frac{1}{2}\right) \cdot \log \left(re \cdot re + im \cdot im\right) \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(\color{blue}{{\log 10}^{-1}} \cdot \frac{1}{2}\right) \cdot \log \left(re \cdot re + im \cdot im\right) \]

   13. lower-log.f64 49.7

       \[\leadsto \left({\log 10}^{-1} \cdot 0.5\right) \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)} \]

   14. lift-+.f64 N/A

       \[\leadsto \left({\log 10}^{-1} \cdot \frac{1}{2}\right) \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)} \]

   15. +-commutative N/A

       \[\leadsto \left({\log 10}^{-1} \cdot \frac{1}{2}\right) \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \left({\log 10}^{-1} \cdot \frac{1}{2}\right) \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right) \]

   17. lower-fma.f64 49.7

       \[\leadsto \left({\log 10}^{-1} \cdot 0.5\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)} \]

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{\left({\log 10}^{-1} \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\log 10}^{-1} \cdot \frac{1}{2}\right)} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {\log 10}^{-1}\right)} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{{\log 10}^{-1}}\right) \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{\log 10}}\right) \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\log 10}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   6. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\mathsf{neg}\left(\log 10\right)}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{\mathsf{neg}\left(\log 10\right)} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{\mathsf{neg}\left(\log 10\right)}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   9. lift-log.f64 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\mathsf{neg}\left(\color{blue}{\log 10}\right)} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   10. neg-log N/A

       \[\leadsto \frac{\frac{-1}{2}}{\color{blue}{\log \left(\frac{1}{10}\right)}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   11. lower-log.f64 N/A

       \[\leadsto \frac{\frac{-1}{2}}{\color{blue}{\log \left(\frac{1}{10}\right)}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

   12. metadata-eval 50.0

       \[\leadsto \frac{-0.5}{\log \color{blue}{0.1}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

6. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{-0.5}{\log 0.1}} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \]

7. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \left(\frac{re \cdot re}{\color{blue}{im \cdot im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \]

   4. times-frac N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \left(\color{blue}{\frac{re}{im} \cdot \frac{re}{im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \]

   8. log-rec N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right) \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{2}}{\log \frac{1}{10}} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right) \]

   13. lower-log.f64 22.2

       \[\leadsto \frac{-0.5}{\log 0.1} \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right) \]

9. Applied rewrites 22.2%

   \[\leadsto \frac{-0.5}{\log 0.1} \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)} \]

10. Final simplification 22.2%

    \[\leadsto \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \log im \cdot 2\right) \cdot \frac{-0.5}{\log 0.1} \]
11. Add Preprocessing

Alternative 4: 26.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\frac{-1}{\log im}}}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (/ -1.0 (/ -1.0 (log im))) (log 10.0)))

C

double code(double re, double im) {
	return (-1.0 / (-1.0 / log(im))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-1.0d0) / ((-1.0d0) / log(im))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return (-1.0 / (-1.0 / Math.log(im))) / Math.log(10.0);
}

Python

def code(re, im):
	return (-1.0 / (-1.0 / math.log(im))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(Float64(-1.0 / Float64(-1.0 / log(im))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (-1.0 / (-1.0 / log(im))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[(-1.0 / N[(-1.0 / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Step-by-step derivation

   1. lift-hypot.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   3. +-commutative N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}{\log 10} \]

   5. lift-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}{\log 10} \]

   6. remove-double-div N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

   7. unpow-1 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}}}{\log 10} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}}}{\log 10} \]

   9. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}}{\log 10} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{-1}}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}{\log 10} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{\mathsf{neg}\left({\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}\right)}}}{\log 10} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}}\right)}}{\log 10} \]

   13. unpow-1 N/A

       \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}\right)}}{\log 10} \]

   14. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{\frac{\color{blue}{-1}}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}{\log 10} \]

   16. lower-/.f64 99.1

       \[\leadsto \frac{\frac{-1}{\color{blue}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

6. Applied rewrites 99.1%

   \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]

7. Taylor expanded in re around 0

   \[\leadsto \frac{\frac{-1}{\frac{-1}{\color{blue}{\log im}}}}{\log 10} \]
8. Step-by-step derivation

   1. lower-log.f64 23.9

      \[\leadsto \frac{\frac{-1}{\frac{-1}{\color{blue}{\log im}}}}{\log 10} \]

9. Applied rewrites 23.9%

   \[\leadsto \frac{\frac{-1}{\frac{-1}{\color{blue}{\log im}}}}{\log 10} \]
10. Add Preprocessing

Alternative 5: 26.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

double code(double re, double im) {
	return log(im) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(im) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(im) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 23.9

      \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

5. Applied rewrites 23.9%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))