math.log10 on complex, real part

Percentage Accurate: 50.7% → 99.7%

Time: 10.3s

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: 50.7% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (pow (pow (log 10.0) -0.5) 2.0))))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), pow(pow(log(10.0), -0.5), 2.0)));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(Math.hypot(re, im), Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0)));
}

Python

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), math.pow(math.pow(math.log(10.0), -0.5), 2.0)))

Julia

function code(re, im)
	return log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)));
end

Wolfram

code[re_, im_] := N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.1%

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

   1. +-commutative 57.1%

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

   2. +-commutative 57.1%

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

   3. sqr-neg 57.1%

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

   4. sqr-neg 57.1%

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

   5. sqr-neg 57.1%

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

   6. sqr-neg 57.1%

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

   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. add-log-exp 99.0%

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

   2. div-inv 98.5%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 98.9%

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

   7. metadata-eval 98.9%

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

6. Applied egg-rr 98.9%

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

   1. metadata-eval 98.9%

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

   2. metadata-eval 98.9%

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

   3. neg-log 98.5%

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

   4. frac-2neg 98.5%

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

   5. inv-pow 98.5%

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

   6. metadata-eval 98.5%

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

   7. pow-prod-up 99.7%

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

   8. pow2 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]
9. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 10}\right)}^{3}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (cbrt (pow (/ (log (hypot im re)) (log 10.0)) 3.0)))

C

double code(double re, double im) {
	return cbrt(pow((log(hypot(im, re)) / log(10.0)), 3.0));
}

Java

public static double code(double re, double im) {
	return Math.cbrt(Math.pow((Math.log(Math.hypot(im, re)) / Math.log(10.0)), 3.0));
}

Julia

function code(re, im)
	return cbrt((Float64(log(hypot(im, re)) / log(10.0)) ^ 3.0))
end

Wolfram

code[re_, im_] := N[Power[N[Power[N[(N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 57.1%

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

   1. +-commutative 57.1%

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

   2. +-commutative 57.1%

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

   3. sqr-neg 57.1%

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

   4. sqr-neg 57.1%

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

   5. sqr-neg 57.1%

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

   6. sqr-neg 57.1%

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

   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. add-cbrt-cube 99.0%

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

   2. add-cbrt-cube 97.9%

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

   3. cbrt-undiv 99.0%

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

   4. pow1/3 73.3%

      \[\leadsto \color{blue}{{\left(\frac{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{0.3333333333333333}} \]

   5. pow3 73.4%

      \[\leadsto {\left(\frac{\color{blue}{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}\right)}^{0.3333333333333333} \]

   6. pow3 73.4%

      \[\leadsto {\left(\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{\color{blue}{{\log 10}^{3}}}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 73.4%

   \[\leadsto \color{blue}{{\left(\frac{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}{{\log 10}^{3}}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. unpow1/3 99.0%

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

   2. unpow3 99.0%

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

   3. unpow3 99.0%

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

   4. times-frac 99.0%

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

   5. times-frac 99.0%

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

   6. unpow2 99.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)}^{2}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

   7. pow-plus 99.1%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)}^{\left(2 + 1\right)}}} \]

   8. hypot-undefine 57.1%

      \[\leadsto \sqrt[3]{{\left(\frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   9. unpow2 57.1%

      \[\leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt{\color{blue}{{re}^{2}} + im \cdot im}\right)}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   10. unpow2 57.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt{{re}^{2} + \color{blue}{{im}^{2}}}\right)}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   11. +-commutative 57.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt{\color{blue}{{im}^{2} + {re}^{2}}}\right)}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   12. unpow2 57.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt{\color{blue}{im \cdot im} + {re}^{2}}\right)}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   13. unpow2 57.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   14. hypot-define 99.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}{\log 10}\right)}^{\left(2 + 1\right)}} \]

   15. metadata-eval 99.1%

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

8. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 10}\right)}^{3}}} \]
9. 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 57.1%

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

   1. +-commutative 57.1%

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

   2. +-commutative 57.1%

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

   3. sqr-neg 57.1%

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

   4. sqr-neg 57.1%

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

   5. sqr-neg 57.1%

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

   6. sqr-neg 57.1%

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

   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. Add Preprocessing

Alternative 4: 27.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{1}{im}\right)}{-\log 10} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

   1. +-commutative 57.1%

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

   2. +-commutative 57.1%

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

   3. sqr-neg 57.1%

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

   4. sqr-neg 57.1%

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

   5. sqr-neg 57.1%

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

   6. sqr-neg 57.1%

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

   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 im around inf 23.4%

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

6. Final simplification 23.4%

   \[\leadsto \frac{\log \left(\frac{1}{im}\right)}{-\log 10} \]
7. Add Preprocessing

Alternative 5: 27.2% 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 57.1%

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

   1. +-commutative 57.1%

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

   2. +-commutative 57.1%

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

   3. sqr-neg 57.1%

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

   4. sqr-neg 57.1%

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

   5. sqr-neg 57.1%

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

   6. sqr-neg 57.1%

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

   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 23.4%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))