math.log10 on complex, real part

Percentage Accurate: 50.9% → 99.5%

Time: 9.1s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right) \cdot \frac{-3}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.3333333333333333 (log (hypot im re))) (/ -3.0 (log 0.1))))

C

double code(double re, double im) {
	return (0.3333333333333333 * log(hypot(im, re))) * (-3.0 / log(0.1));
}

Java

public static double code(double re, double im) {
	return (0.3333333333333333 * Math.log(Math.hypot(im, re))) * (-3.0 / Math.log(0.1));
}

Python

def code(re, im):
	return (0.3333333333333333 * math.log(math.hypot(im, re))) * (-3.0 / math.log(0.1))

Julia

function code(re, im)
	return Float64(Float64(0.3333333333333333 * log(hypot(im, re))) * Float64(-3.0 / log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.3333333333333333 * log(hypot(im, re))) * (-3.0 / log(0.1));
end

Wolfram

code[re_, im_] := N[(N[(0.3333333333333333 * N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $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%

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

   5. 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%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log 0.1}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 99.3%

       \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \color{blue}{\frac{3 \cdot -1}{\log 0.1}} \]

    2. metadata-eval 99.3%

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

12. Simplified 99.3%

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

    1. pow1/3 99.6%

       \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(im, re\right)\right)}^{0.3333333333333333}\right)} \cdot \frac{-3}{\log 0.1} \]

    2. log-pow 99.4%

       \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \cdot \frac{-3}{\log 0.1} \]

14. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \cdot \frac{-3}{\log 0.1} \]
15. Add Preprocessing

Alternative 2: 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%

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

   5. 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 3: 27.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-3}{\log 0.1} \cdot \left(0.3333333333333333 \cdot \log im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (/ -3.0 (log 0.1)) (* 0.3333333333333333 (log im))))

C

double code(double re, double im) {
	return (-3.0 / log(0.1)) * (0.3333333333333333 * log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-3.0d0) / log(0.1d0)) * (0.3333333333333333d0 * log(im))
end function

Java

public static double code(double re, double im) {
	return (-3.0 / Math.log(0.1)) * (0.3333333333333333 * Math.log(im));
}

Python

def code(re, im):
	return (-3.0 / math.log(0.1)) * (0.3333333333333333 * math.log(im))

Julia

function code(re, im)
	return Float64(Float64(-3.0 / log(0.1)) * Float64(0.3333333333333333 * log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (-3.0 / log(0.1)) * (0.3333333333333333 * log(im));
end

Wolfram

code[re_, im_] := N[(N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Log[im], $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%

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

   5. 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%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log 0.1}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 99.3%

       \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(im, re\right)}\right) \cdot \color{blue}{\frac{3 \cdot -1}{\log 0.1}} \]

    2. metadata-eval 99.3%

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

12. Simplified 99.3%

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

    1. pow1/3 99.6%

       \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(im, re\right)\right)}^{0.3333333333333333}\right)} \cdot \frac{-3}{\log 0.1} \]

    2. log-pow 99.4%

       \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \cdot \frac{-3}{\log 0.1} \]

14. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \cdot \frac{-3}{\log 0.1} \]

15. Taylor expanded in re around 0 25.2%

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log im\right)} \cdot \frac{-3}{\log 0.1} \]

16. Final simplification 25.2%

    \[\leadsto \frac{-3}{\log 0.1} \cdot \left(0.3333333333333333 \cdot \log im\right) \]
17. 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%

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

   5. 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)))