math.log10 on complex, real part

Percentage Accurate: 51.5% → 99.0%

Time: 6.6s

Alternatives: 6

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: 51.5% 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

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ -1.0 (/ (log 0.1) (log (hypot im re)))))

C

double code(double re, double im) {
	return -1.0 / (log(0.1) / log(hypot(im, re)));
}

Java

public static double code(double re, double im) {
	return -1.0 / (Math.log(0.1) / Math.log(Math.hypot(im, re)));
}

Python

def code(re, im):
	return -1.0 / (math.log(0.1) / math.log(math.hypot(im, re)))

Julia

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / log(hypot(im, re))))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 / (log(0.1) / log(hypot(im, re)));
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. lower-log.f64 N/A

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

   11. metadata-eval 53.4

       \[\leadsto \frac{-1}{\frac{\log \color{blue}{0.1}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \]

   12. lift-sqrt.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-hypot.f64 99.1

       \[\leadsto \frac{-1}{\frac{\log 0.1}{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}} \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}} \]
5. 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 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.0

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

4. Applied rewrites 99.0%

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

Alternative 3: 24.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.0

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 0.1}} \]
5. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-hypot.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. log-pow N/A

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

   10. distribute-lft-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 53.4

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

   14. lift-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 53.4

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

6. Applied rewrites 53.4%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 26.3

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

9. Applied rewrites 26.3%

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

10. Taylor expanded in im around inf

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. unpow2 N/A

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

    4. times-frac N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. log-rec N/A

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

    9. mul-1-neg N/A

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

    10. associate-*r* N/A

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

    11. metadata-eval N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lower-log.f64 26.3

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

12. Applied rewrites 26.3%

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

Alternative 4: 26.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ -1.0 (/ (log 0.1) (log im))))

C

double code(double re, double im) {
	return -1.0 / (log(0.1) / log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / (log(0.1d0) / log(im))
end function

Java

public static double code(double re, double im) {
	return -1.0 / (Math.log(0.1) / Math.log(im));
}

Python

def code(re, im):
	return -1.0 / (math.log(0.1) / math.log(im))

Julia

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 / (log(0.1) / log(im));
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 28.3

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

5. Applied rewrites 28.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-log.f64 N/A

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

   7. neg-log N/A

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

   8. metadata-eval N/A

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

   9. lift-log.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 28.3

       \[\leadsto \frac{-1}{\color{blue}{\frac{\log 0.1}{\log im}}} \]

7. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log im}}} \]
8. Add Preprocessing

Alternative 5: 26.5% 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 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 28.3

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

5. Applied rewrites 28.3%

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

Alternative 6: 3.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.5d0) * ((re / im) * (re / im))) / log(0.1d0)
end function

Java

public static double code(double re, double im) {
	return (-0.5 * ((re / im) * (re / im))) / Math.log(0.1);
}

Python

def code(re, im):
	return (-0.5 * ((re / im) * (re / im))) / math.log(0.1)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (-0.5 * ((re / im) * (re / im))) / log(0.1);
end

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.0

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 0.1}} \]
5. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-hypot.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. log-pow N/A

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

   10. distribute-lft-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 53.4

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

   14. lift-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 53.4

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

6. Applied rewrites 53.4%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 26.3

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

9. Applied rewrites 26.3%

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

10. Taylor expanded in re around inf

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

12. Applied rewrites 3.4%

    \[\leadsto \frac{-0.5 \cdot \left(\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}\right)}{\log 0.1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))