math.log10 on complex, real part

Percentage Accurate: 50.8% → 99.5%

Time: 14.3s

Alternatives: 7

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: 50.8% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log (hypot im re)) (/ (pow (log 10.0) -0.5) (sqrt (log 10.0)))))

C

double code(double re, double im) {
	return log(hypot(im, re)) * (pow(log(10.0), -0.5) / sqrt(log(10.0)));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(im, re)) * (Math.pow(Math.log(10.0), -0.5) / Math.sqrt(Math.log(10.0)));
}

Python

def code(re, im):
	return math.log(math.hypot(im, re)) * (math.pow(math.log(10.0), -0.5) / math.sqrt(math.log(10.0)))

Julia

function code(re, im)
	return Float64(log(hypot(im, re)) * Float64((log(10.0) ^ -0.5) / sqrt(log(10.0))))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(im, re)) * ((log(10.0) ^ -0.5) / sqrt(log(10.0)));
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-flip 99.2%

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

   6. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/l* 99.6%

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

   4. hypot-undefine 51.4%

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

   5. unpow2 51.4%

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

   6. unpow2 51.4%

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

   7. +-commutative 51.4%

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

   8. unpow2 51.4%

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

   9. unpow2 51.4%

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

   10. hypot-define 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 27.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow im (log1p (expm1 (/ 1.0 (log 10.0)))))))

C

double code(double re, double im) {
	return log(pow(im, log1p(expm1((1.0 / log(10.0))))));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(im, Math.log1p(Math.expm1((1.0 / Math.log(10.0))))));
}

Python

def code(re, im):
	return math.log(math.pow(im, math.log1p(math.expm1((1.0 / math.log(10.0))))))

Julia

function code(re, im)
	return log((im ^ log1p(expm1(Float64(1.0 / log(10.0))))))
end

Wolfram

code[re_, im_] := N[Log[N[Power[im, N[Log[1 + N[(Exp[N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 24.5%

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

   1. div-inv 24.4%

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

   2. inv-pow 24.4%

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

   3. metadata-eval 24.4%

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

   4. pow-div 24.6%

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

   5. pow1/2 24.6%

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

   6. add-log-exp 24.6%

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

   7. exp-to-pow 24.7%

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

   8. pow1/2 24.7%

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

   9. pow-div 24.4%

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

   10. metadata-eval 24.4%

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

   11. inv-pow 24.4%

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

   12. frac-2neg 24.4%

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

   13. metadata-eval 24.4%

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

   14. neg-log 24.5%

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

   15. metadata-eval 24.5%

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

7. Applied egg-rr 24.5%

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

   1. metadata-eval 24.5%

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

   2. metadata-eval 24.5%

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

   3. neg-log 24.4%

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

   4. frac-2neg 24.4%

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

   5. log1p-expm1-u 24.7%

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

9. Applied egg-rr 24.7%

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

10. Final simplification 24.7%

    \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}\right) \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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. clear-num 99.1%

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

   2. inv-pow 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-flip 99.2%

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

   6. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/l* 99.6%

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

   4. hypot-undefine 51.4%

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

   5. unpow2 51.4%

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

   6. unpow2 51.4%

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

   7. +-commutative 51.4%

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

   8. unpow2 51.4%

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

   9. unpow2 51.4%

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

   10. hypot-define 99.6%

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

8. Simplified 99.6%

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

   1. add-log-exp 99.6%

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

   2. exp-to-pow 99.7%

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

   3. hypot-undefine 51.4%

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

   4. +-commutative 51.4%

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

   5. hypot-undefine 99.7%

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

   6. exp-to-pow 99.6%

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

   7. add-log-exp 99.6%

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

   8. *-commutative 99.6%

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

   9. pow1/2 99.6%

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

   10. pow-div 98.6%

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

   11. metadata-eval 98.6%

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

   12. inv-pow 98.6%

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

   13. associate-/r/ 99.1%

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

   14. div-inv 99.0%

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

   15. div-inv 99.1%

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

   16. hypot-undefine 51.1%

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

   17. +-commutative 51.1%

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

10. Applied egg-rr 99.1%

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

11. Final simplification 99.1%

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

Alternative 5: 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 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 6: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 24.5%

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

   1. clear-num 24.5%

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

   2. inv-pow 24.5%

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

7. Applied egg-rr 24.5%

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

   1. unpow-1 24.5%

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

9. Simplified 24.5%

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

10. Final simplification 24.5%

    \[\leadsto \frac{1}{\frac{\log 10}{\log im}} \]
11. Add Preprocessing

Alternative 7: 27.7% 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 51.1%

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

   1. +-commutative 51.1%

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

   2. +-commutative 51.1%

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

   3. sqr-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sqr-neg 51.1%

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

   6. sqr-neg 51.1%

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

   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 24.5%

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

6. Final simplification 24.5%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))