math.log10 on complex, real part

Percentage Accurate: 50.9% → 99.5%

Time: 10.9s

Alternatives: 4

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: 50.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. sqr-neg 50.7%

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

   4. sqr-neg 50.7%

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

   5. sqr-neg 50.7%

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

   6. sqr-neg 50.7%

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

   7. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

   3. add-sqr-sqrt 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. times-frac 98.7%

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

   6. unpow-prod-down 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. /-rgt-identity 99.1%

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

   3. associate-*l/ 98.8%

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

   4. *-lft-identity 98.8%

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

   5. unpow-1 98.8%

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

   6. associate-/r/ 99.2%

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

   7. hypot-undefine 50.8%

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

   8. unpow2 50.8%

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

   9. unpow2 50.8%

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

   10. +-commutative 50.8%

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

   11. unpow2 50.8%

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

   12. unpow2 50.8%

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

   13. hypot-define 99.2%

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

8. Simplified 99.2%

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

   1. associate-/l* 99.1%

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

   2. pow1/2 99.1%

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

   3. pow-flip 99.1%

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

   4. metadata-eval 99.1%

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

10. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.2%

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

    2. *-commutative 99.2%

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

    3. *-lft-identity 99.2%

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

    4. times-frac 99.6%

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

    5. /-rgt-identity 99.6%

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

12. Simplified 99.6%

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

    1. add-sqr-sqrt 99.6%

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

    2. sqrt-unprod 99.6%

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

    3. pow1/2 99.6%

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

    4. pow-div 99.6%

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

    5. metadata-eval 99.6%

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

    6. inv-pow 99.6%

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

    7. pow1/2 99.6%

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

    8. pow-div 98.6%

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

    9. metadata-eval 98.6%

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

    10. inv-pow 98.6%

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

    11. frac-times 98.6%

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

    12. metadata-eval 98.6%

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

    13. pow2 98.6%

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

14. Applied egg-rr 98.6%

    \[\leadsto \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{{\log 10}^{2}}}} \]
15. Step-by-step derivation

    1. unpow2 98.6%

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

    2. associate-/r* 98.6%

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

    3. *-rgt-identity 98.6%

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

    4. associate-*r/ 98.6%

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

    5. unpow-1 98.6%

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

    6. unpow-1 98.6%

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

    7. pow-sqr 99.6%

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

    8. metadata-eval 99.6%

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

16. Simplified 99.6%

    \[\leadsto \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \color{blue}{\sqrt{{\log 10}^{-2}}} \]
17. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \frac{-3}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log (cbrt (hypot im re))) (/ -3.0 (log 0.1))))

C

double code(double re, double im) {
	return log(cbrt(hypot(im, re))) * (-3.0 / log(0.1));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.cbrt(Math.hypot(im, re))) * (-3.0 / Math.log(0.1));
}

Julia

function code(re, im)
	return Float64(log(cbrt(hypot(im, re))) * Float64(-3.0 / log(0.1)))
end

Wolfram

code[re_, im_] := N[(N[Log[N[Power[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. sqr-neg 50.7%

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

   4. sqr-neg 50.7%

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

   5. sqr-neg 50.7%

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

   6. sqr-neg 50.7%

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

   7. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

   3. add-sqr-sqrt 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. times-frac 98.7%

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

   6. unpow-prod-down 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. /-rgt-identity 99.1%

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

   3. associate-*l/ 98.8%

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

   4. *-lft-identity 98.8%

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

   5. unpow-1 98.8%

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

   6. associate-/r/ 99.2%

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

   7. hypot-undefine 50.8%

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

   8. unpow2 50.8%

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

   9. unpow2 50.8%

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

   10. +-commutative 50.8%

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

   11. unpow2 50.8%

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

   12. unpow2 50.8%

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

   13. hypot-define 99.2%

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

8. Simplified 99.2%

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

   1. add-log-exp 99.2%

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

   2. *-commutative 99.2%

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

   3. exp-to-pow 99.2%

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

   4. hypot-undefine 50.8%

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

   5. +-commutative 50.8%

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

   6. hypot-undefine 99.2%

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

   7. exp-to-pow 99.2%

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

   8. div-inv 98.8%

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

   9. add-log-exp 98.8%

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

   10. associate-/r* 99.1%

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

   11. add-sqr-sqrt 99.1%

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

   12. div-inv 98.6%

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

   13. add-log-exp 98.5%

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

   14. exp-to-pow 98.5%

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

10. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{3 \cdot \frac{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right)}{\log 10}} \]
11. Step-by-step derivation

    1. *-commutative 99.1%

       \[\leadsto \color{blue}{\frac{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right)}{\log 10} \cdot 3} \]

    2. associate-*l/ 99.1%

       \[\leadsto \color{blue}{\frac{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{\log 10}} \]

    3. associate-*r/ 98.9%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \frac{3}{\log 10}} \]

12. Simplified 98.9%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \frac{3}{\log 10}} \]
13. Step-by-step derivation

    1. associate-*r/ 99.1%

       \[\leadsto \color{blue}{\frac{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{\log 10}} \]

    2. frac-2neg 99.1%

       \[\leadsto \color{blue}{\frac{-\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{-\log 10}} \]

    3. neg-log 99.0%

       \[\leadsto \frac{-\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

    4. metadata-eval 99.0%

       \[\leadsto \frac{-\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{\log \color{blue}{0.1}} \]

14. Applied egg-rr 99.0%

    \[\leadsto \color{blue}{\frac{-\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot 3}{\log 0.1}} \]
15. Step-by-step derivation

    1. distribute-rgt-neg-in 99.0%

       \[\leadsto \frac{\color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \left(-3\right)}}{\log 0.1} \]

    2. metadata-eval 99.0%

       \[\leadsto \frac{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \color{blue}{-3}}{\log 0.1} \]

    3. associate-/l* 99.3%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \frac{-3}{\log 0.1}} \]

16. Simplified 99.3%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \frac{-3}{\log 0.1}} \]
17. Add Preprocessing

Alternative 3: 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 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. sqr-neg 50.7%

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

   4. sqr-neg 50.7%

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

   5. sqr-neg 50.7%

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

   6. sqr-neg 50.7%

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

   7. hypot-define 99.1%

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

3. Simplified 99.1%

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

Alternative 4: 27.5% 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 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. sqr-neg 50.7%

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

   4. sqr-neg 50.7%

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

   5. sqr-neg 50.7%

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

   6. sqr-neg 50.7%

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

   7. hypot-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in re around 0 25.2%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))