math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.5%

Time: 12.9s

Alternatives: 6

Speedup: 1.0×

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: 51.4% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\sqrt{\log 10}}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (pow (/ 1.0 (sqrt (log 10.0))) 2.0) (log (hypot im re))))

C

double code(double re, double im) {
	return pow((1.0 / sqrt(log(10.0))), 2.0) * log(hypot(im, re));
}

Java

public static double code(double re, double im) {
	return Math.pow((1.0 / Math.sqrt(Math.log(10.0))), 2.0) * Math.log(Math.hypot(im, re));
}

Python

def code(re, im):
	return math.pow((1.0 / math.sqrt(math.log(10.0))), 2.0) * math.log(math.hypot(im, re))

Julia

function code(re, im)
	return Float64((Float64(1.0 / sqrt(log(10.0))) ^ 2.0) * log(hypot(im, re)))
end

MATLAB

function tmp = code(re, im)
	tmp = ((1.0 / sqrt(log(10.0))) ^ 2.0) * log(hypot(im, re));
end

Wolfram

code[re_, im_] := N[(N[Power[N[(1.0 / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\log 10}^{-0.5} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]
7. Step-by-step derivation

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. associate-/l* 99.5%

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

   4. hypot-undefine 54.1%

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

   5. unpow2 54.1%

      \[\leadsto \log \left(\sqrt{\color{blue}{{re}^{2}} + im \cdot im}\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \]

   6. unpow2 54.1%

      \[\leadsto \log \left(\sqrt{{re}^{2} + \color{blue}{{im}^{2}}}\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \]

   7. +-commutative 54.1%

      \[\leadsto \log \left(\sqrt{\color{blue}{{im}^{2} + {re}^{2}}}\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \]

   8. unpow2 54.1%

      \[\leadsto \log \left(\sqrt{\color{blue}{im \cdot im} + {re}^{2}}\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \]

   9. unpow2 54.1%

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

   10. hypot-define 99.5%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}} \]
9. Step-by-step derivation

   1. add-log-exp 99.5%

      \[\leadsto \color{blue}{\log \left(e^{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}}\right)} \]

   2. exp-to-pow 99.8%

      \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(im, re\right)\right)}^{\left(\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}\right)}\right)} \]

   3. hypot-undefine 54.2%

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

   4. +-commutative 54.2%

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

   5. hypot-undefine 99.8%

      \[\leadsto \log \left({\color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}^{\left(\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}\right)}\right) \]

   6. exp-to-pow 99.5%

      \[\leadsto \log \color{blue}{\left(e^{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}}\right)} \]

   7. add-log-exp 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-div 98.5%

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

   10. metadata-eval 98.5%

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

   11. inv-pow 98.5%

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

   12. div-inv 99.0%

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

   13. frac-2neg 99.0%

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

   14. hypot-undefine 53.8%

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

   15. +-commutative 53.8%

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

   16. hypot-undefine 99.0%

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

10. Applied egg-rr 99.1%

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

    1. neg-mul-1 99.1%

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

    2. associate-*l/ 99.1%

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

    3. add-sqr-sqrt 99.5%

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

    4. associate-*l* 99.4%

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

    5. metadata-eval 99.4%

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

    6. metadata-eval 99.4%

       \[\leadsto \sqrt{\frac{-1}{\log \color{blue}{\left(\frac{1}{10}\right)}}} \cdot \left(\sqrt{\frac{-1}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    7. neg-log 99.4%

       \[\leadsto \sqrt{\frac{-1}{\color{blue}{-\log 10}}} \cdot \left(\sqrt{\frac{-1}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    8. frac-2neg 99.4%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\log 10}}} \cdot \left(\sqrt{\frac{-1}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    9. sqrt-div 99.4%

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{-1}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    10. metadata-eval 99.4%

        \[\leadsto \frac{\color{blue}{1}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{-1}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    11. metadata-eval 99.4%

        \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{\color{blue}{-1}}{\log 0.1}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \]

    12. metadata-eval 99.4%

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

    13. neg-log 99.4%

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

    14. frac-2neg 99.4%

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

    15. sqrt-div 99.4%

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

    16. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r* 99.5%

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

    2. unpow2 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

    \[\leadsto {\left(\frac{1}{\sqrt{\log 10}}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \]
16. Add Preprocessing

Alternative 2: 26.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(-0.9\right)}\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow im (log1p (expm1 (/ -1.0 (log1p -0.9)))))))

C

double code(double re, double im) {
	return log(pow(im, log1p(expm1((-1.0 / log1p(-0.9))))));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(im, Math.log1p(Math.expm1((-1.0 / Math.log1p(-0.9))))));
}

Python

def code(re, im):
	return math.log(math.pow(im, math.log1p(math.expm1((-1.0 / math.log1p(-0.9))))))

Julia

function code(re, im)
	return log((im ^ log1p(expm1(Float64(-1.0 / log1p(-0.9))))))
end

Wolfram

code[re_, im_] := N[Log[N[Power[im, N[Log[1 + N[(Exp[N[(-1.0 / N[Log[1 + -0.9], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 24.3%

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

   1. div-inv 24.2%

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

   2. inv-pow 24.2%

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

   3. metadata-eval 24.2%

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

   4. pow-div 24.4%

      \[\leadsto \log im \cdot \color{blue}{\frac{{\log 10}^{-0.5}}{{\log 10}^{0.5}}} \]

   5. pow1/2 24.4%

      \[\leadsto \log im \cdot \frac{{\log 10}^{-0.5}}{\color{blue}{\sqrt{\log 10}}} \]

   6. add-log-exp 24.4%

      \[\leadsto \color{blue}{\log \left(e^{\log im \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}}\right)} \]

   7. exp-to-pow 24.5%

      \[\leadsto \log \color{blue}{\left({im}^{\left(\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}}\right)}\right)} \]

   8. pow1/2 24.5%

      \[\leadsto \log \left({im}^{\left(\frac{{\log 10}^{-0.5}}{\color{blue}{{\log 10}^{0.5}}}\right)}\right) \]

   9. pow-div 24.2%

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

   10. metadata-eval 24.2%

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

   11. inv-pow 24.2%

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

   12. frac-2neg 24.2%

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

   13. metadata-eval 24.2%

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

   14. neg-log 24.3%

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

   15. metadata-eval 24.3%

       \[\leadsto \log \left({im}^{\left(\frac{-1}{\log \color{blue}{0.1}}\right)}\right) \]

7. Applied egg-rr 24.3%

   \[\leadsto \color{blue}{\log \left({im}^{\left(\frac{-1}{\log 0.1}\right)}\right)} \]
8. Step-by-step derivation

   1. metadata-eval 24.3%

      \[\leadsto \log \left({im}^{\left(\frac{\color{blue}{-1}}{\log 0.1}\right)}\right) \]

   2. metadata-eval 24.3%

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

   3. neg-log 24.2%

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

   4. frac-2neg 24.2%

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

   5. log1p-expm1-u 24.5%

      \[\leadsto \log \left({im}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}}\right) \]

   6. frac-2neg 24.5%

      \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{-1}{-\log 10}}\right)\right)\right)}\right) \]

   7. metadata-eval 24.5%

      \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-1}}{-\log 10}\right)\right)\right)}\right) \]

   8. log1p-expm1-u 24.5%

      \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\log 10\right)\right)}}\right)\right)\right)}\right) \]

   9. neg-log 24.5%

      \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log \left(\frac{1}{10}\right)}\right)\right)}\right)\right)\right)}\right) \]

   10. metadata-eval 24.5%

       \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{0.1}\right)\right)}\right)\right)\right)}\right) \]

   11. expm1-undefine 24.5%

       \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(\color{blue}{e^{\log 0.1} - 1}\right)}\right)\right)\right)}\right) \]

   12. rem-exp-log 24.5%

       \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(\color{blue}{0.1} - 1\right)}\right)\right)\right)}\right) \]

   13. metadata-eval 24.5%

       \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(\color{blue}{-0.9}\right)}\right)\right)\right)}\right) \]

9. Applied egg-rr 24.5%

   \[\leadsto \log \left({im}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(-0.9\right)}\right)\right)\right)}}\right) \]

10. Final simplification 24.5%

    \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{\mathsf{log1p}\left(-0.9\right)}\right)\right)\right)}\right) \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{-1}{\log 0.1}} \]
7. Step-by-step derivation

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. distribute-neg-frac2 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{-\log 0.1} \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× 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 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 5: 26.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 24.3%

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

   1. clear-num 24.3%

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

   2. associate-/r/ 24.2%

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

   3. frac-2neg 24.2%

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

   4. metadata-eval 24.2%

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

   5. neg-log 24.3%

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

   6. metadata-eval 24.3%

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

7. Applied egg-rr 24.3%

   \[\leadsto \color{blue}{\frac{-1}{\log 0.1} \cdot \log im} \]
8. Step-by-step derivation

   1. associate-*l/ 24.4%

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

   2. neg-mul-1 24.4%

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

9. Simplified 24.4%

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

10. Final simplification 24.4%

    \[\leadsto \frac{\log im}{-\log 0.1} \]
11. Add Preprocessing

Alternative 6: 26.4% accurate, 1.5× 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 53.8%

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

   1. +-commutative 53.8%

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

   2. +-commutative 53.8%

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

   3. sqr-neg 53.8%

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

   4. sqr-neg 53.8%

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

   5. sqr-neg 53.8%

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

   6. sqr-neg 53.8%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 24.3%

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

6. Final simplification 24.3%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))