math.log10 on complex, real part

Percentage Accurate: 51.0% → 99.6%

Time: 10.5s

Alternatives: 5

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: 51.0% 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: 99.6% 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 49.8%

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

   1. +-commutative 49.8%

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

   2. +-commutative 49.8%

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

   3. sqr-neg 49.8%

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

   4. sqr-neg 49.8%

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

   5. sqr-neg 49.8%

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

   6. sqr-neg 49.8%

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

   7. hypot-define 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. add-sqr-sqrt 98.9%

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

   3. times-frac 99.0%

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

   4. pow1/2 99.0%

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

   5. pow-flip 99.0%

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

   6. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   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 50.1%

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

   5. unpow2 50.1%

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

   6. unpow2 50.1%

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

   7. +-commutative 50.1%

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

   8. unpow2 50.1%

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

   9. unpow2 50.1%

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

   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. Final simplification 99.5%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

   1. +-commutative 49.8%

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

   2. +-commutative 49.8%

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

   3. sqr-neg 49.8%

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

   4. sqr-neg 49.8%

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

   5. sqr-neg 49.8%

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

   6. sqr-neg 49.8%

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

   7. hypot-define 98.9%

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

3. Simplified 98.9%

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

   1. add-log-exp 98.9%

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

   2. add-sqr-sqrt 98.9%

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

   3. log-prod 98.9%

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

   4. div-inv 98.7%

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

   5. exp-to-pow 98.7%

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

   6. frac-2neg 98.7%

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

   7. metadata-eval 98.7%

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

   8. neg-log 99.4%

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

   9. metadata-eval 99.4%

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

   10. div-inv 99.4%

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

   11. exp-to-pow 99.5%

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

   12. frac-2neg 99.5%

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

   13. metadata-eval 99.5%

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

   14. neg-log 99.1%

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

6. Applied egg-rr 99.1%

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

   1. count-2 99.1%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1}\right)}}\right)} \]

   2. sqr-pow 99.1%

      \[\leadsto 2 \cdot \log \left(\sqrt{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}}}\right) \]

   3. rem-sqrt-square 99.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(\left|{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}\right|\right)} \]

   4. sqr-pow 99.1%

      \[\leadsto 2 \cdot \log \left(\left|\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{\frac{-1}{\log 0.1}}{2}}{2}\right)} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{\frac{-1}{\log 0.1}}{2}}{2}\right)}}\right|\right) \]

   5. fabs-sqr 99.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{\frac{-1}{\log 0.1}}{2}}{2}\right)} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{\frac{-1}{\log 0.1}}{2}}{2}\right)}\right)} \]

   6. sqr-pow 99.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}\right)} \]

   7. count-2 99.1%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}\right) + \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}\right)} \]

   8. log-prod 99.1%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{-1}{\log 0.1}}{2}\right)}\right)} \]

   9. sqr-pow 99.1%

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

   10. log-pow 99.1%

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

   11. associate-*l/ 99.1%

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

   12. neg-mul-1 99.1%

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

   13. distribute-neg-frac 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{-\log 0.1} \]
10. 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 49.8%

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

   1. +-commutative 49.8%

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

   2. +-commutative 49.8%

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

   3. sqr-neg 49.8%

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

   4. sqr-neg 49.8%

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

   5. sqr-neg 49.8%

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

   6. sqr-neg 49.8%

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

   7. hypot-define 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 26.8% 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 49.8%

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

   1. +-commutative 49.8%

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

   2. +-commutative 49.8%

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

   3. sqr-neg 49.8%

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

   4. sqr-neg 49.8%

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

   5. sqr-neg 49.8%

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

   6. sqr-neg 49.8%

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

   7. hypot-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in re around 0 28.0%

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

   1. clear-num 28.0%

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

   2. associate-/r/ 27.9%

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

   3. frac-2neg 27.9%

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

   4. metadata-eval 27.9%

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

   5. neg-log 28.0%

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

   6. metadata-eval 28.0%

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

7. Applied egg-rr 28.0%

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

   1. associate-*l/ 28.0%

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

   2. neg-mul-1 28.0%

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

9. Simplified 28.0%

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

10. Final simplification 28.0%

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

Alternative 5: 26.8% 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 49.8%

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

   1. +-commutative 49.8%

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

   2. +-commutative 49.8%

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

   3. sqr-neg 49.8%

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

   4. sqr-neg 49.8%

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

   5. sqr-neg 49.8%

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

   6. sqr-neg 49.8%

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

   7. hypot-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in re around 0 28.0%

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

6. Final simplification 28.0%

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

Reproduce

herbie shell --seed 2024062 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))