math.log10 on complex, real part

Percentage Accurate: 52.9% → 99.6%

Time: 11.1s

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: 52.9% 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: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. hypot-define 99.0%

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

   2. expm1-log1p-u 74.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

4. Applied egg-rr 74.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 99.0%

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

   2. rem-cbrt-cube 98.8%

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

   3. rem-cbrt-cube 97.9%

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

   4. cbrt-div 98.9%

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

   5. pow1/3 73.3%

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

   6. div-inv 73.3%

      \[\leadsto {\color{blue}{\left({\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3} \cdot \frac{1}{{\log 10}^{3}}\right)}}^{0.3333333333333333} \]

   7. unpow-prod-down 73.5%

      \[\leadsto \color{blue}{{\left({\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.3333333333333333}} \]

   8. pow1/3 99.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{3}}} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.3333333333333333} \]

   9. rem-cbrt-cube 99.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.3333333333333333} \]

   10. pow-flip 99.6%

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

   11. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. unpow1/3 99.6%

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

8. Simplified 99.6%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \sqrt[3]{{\log 10}^{-3}}} \]
9. Add Preprocessing

Alternative 2: 27.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\log 10}^{-2}} \cdot \log im \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((log(10.0d0) ** (-2.0d0))) * log(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0 23.7%

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

   1. clear-num 23.7%

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

   2. associate-/r/ 23.6%

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

5. Applied egg-rr 23.6%

   \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log im} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 23.8%

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

   2. sqrt-unprod 23.6%

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

   3. inv-pow 23.6%

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

   4. inv-pow 23.6%

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

   5. pow-prod-up 23.8%

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

   6. metadata-eval 23.8%

      \[\leadsto \sqrt{{\log 10}^{\color{blue}{-2}}} \cdot \log im \]

7. Applied egg-rr 23.8%

   \[\leadsto \color{blue}{\sqrt{{\log 10}^{-2}}} \cdot \log im \]
8. Add Preprocessing

Alternative 3: 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 52.2%

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

   1. frac-2neg 52.2%

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

   2. div-inv 52.0%

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

   3. hypot-define 98.5%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

   3. distribute-neg-frac 99.0%

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

   4. distribute-neg-frac2 99.0%

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

   5. hypot-undefine 52.2%

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

   6. unpow2 52.2%

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

   7. unpow2 52.2%

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

   8. +-commutative 52.2%

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

   9. unpow2 52.2%

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

   10. unpow2 52.2%

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

   11. hypot-undefine 99.0%

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

6. Simplified 99.0%

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

Alternative 4: 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 52.2%

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

   1. +-commutative 52.2%

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

   2. +-commutative 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. sqr-neg 52.2%

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

   6. sqr-neg 52.2%

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

   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 5: 27.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ -1.0 (/ (log 0.1) (log im))))

C

double code(double re, double im) {
	return -1.0 / (log(0.1) / log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / (log(0.1d0) / log(im))
end function

Java

public static double code(double re, double im) {
	return -1.0 / (Math.log(0.1) / Math.log(im));
}

Python

def code(re, im):
	return -1.0 / (math.log(0.1) / math.log(im))

Julia

function code(re, im)
	return Float64(-1.0 / Float64(log(0.1) / log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 / (log(0.1) / log(im));
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0 23.7%

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

   1. add-exp-log 15.3%

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

5. Applied egg-rr 15.3%

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

7. Applied egg-rr 23.7%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log im}}} \]
8. Add Preprocessing

Alternative 6: 27.0% 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 52.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0 23.7%

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

   1. frac-2neg 23.7%

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

   2. div-inv 23.6%

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

   3. neg-log 23.7%

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

   4. metadata-eval 23.7%

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

5. Applied egg-rr 23.7%

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

   1. log-rec 23.7%

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

   2. associate-*r/ 23.7%

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

   3. *-rgt-identity 23.7%

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

   4. log-rec 23.7%

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

   5. distribute-neg-frac 23.7%

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

   6. distribute-neg-frac2 23.7%

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

7. Simplified 23.7%

   \[\leadsto \color{blue}{\frac{\log im}{-\log 0.1}} \]
8. Add Preprocessing

Alternative 7: 27.0% 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 52.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0 23.7%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))