math.log10 on complex, real part

Percentage Accurate: 52.0% → 99.0%

Time: 7.7s

Alternatives: 5

Speedup: 1.5×

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: 52.0% 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: 99.0% accurate, 0.6× speedup

Math

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

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im)
 :precision binary64
 (log (pow im (pow (pow (log 10.0) -0.5) 2.0))))

C

re = abs(re);
im = abs(im);
assert(re < im);
double code(double re, double im) {
	return log(pow(im, pow(pow(log(10.0), -0.5), 2.0)));
}

Fortran

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im ** ((log(10.0d0) ** (-0.5d0)) ** 2.0d0)))
end function

Java

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

Python

re = abs(re)
im = abs(im)
[re, im] = sort([re, im])
def code(re, im):
	return math.log(math.pow(im, math.pow(math.pow(math.log(10.0), -0.5), 2.0)))

Julia

re = abs(re)
im = abs(im)
re, im = sort([re, im])
function code(re, im)
	return log((im ^ ((log(10.0) ^ -0.5) ^ 2.0)))
end

MATLAB

re = abs(re)
im = abs(im)
re, im = num2cell(sort([re, im])){:}
function tmp = code(re, im)
	tmp = log((im ^ ((log(10.0) ^ -0.5) ^ 2.0)));
end

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
code[re_, im_] := N[Log[N[Power[im, 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. hypot-def 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. Taylor expanded in re around 0 28.1%

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

   1. frac-2neg 28.1%

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

   2. div-inv 28.0%

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

   3. neg-log 28.1%

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

   4. metadata-eval 28.1%

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

6. Applied egg-rr 28.1%

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

   1. log-rec 28.1%

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

   2. associate-*r/ 28.1%

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

   3. *-rgt-identity 28.1%

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

   4. log-rec 28.1%

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

8. Simplified 28.1%

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

   1. metadata-eval 28.1%

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

   2. neg-log 28.1%

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

   3. frac-2neg 28.1%

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

   4. rem-cbrt-cube 28.1%

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

   5. add-log-exp 28.1%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{{\left(\frac{\log im}{\log 10}\right)}^{3}}}\right)} \]

   6. rem-cbrt-cube 28.1%

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

   7. div-inv 28.0%

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

   8. inv-pow 28.0%

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

   9. metadata-eval 28.0%

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

   10. pow-prod-up 28.2%

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

   11. exp-to-pow 28.3%

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

   12. pow-prod-up 28.0%

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

   13. metadata-eval 28.0%

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

   14. inv-pow 28.0%

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

   15. frac-2neg 28.0%

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

   16. metadata-eval 28.0%

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

   17. neg-log 28.1%

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

   18. metadata-eval 28.1%

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

10. Applied egg-rr 28.1%

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

    1. metadata-eval 28.1%

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

    2. metadata-eval 28.1%

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

    3. neg-log 28.0%

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

    4. frac-2neg 28.0%

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

    5. inv-pow 28.0%

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

    6. metadata-eval 28.0%

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

    7. pow-sqr 28.3%

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

    8. pow2 28.3%

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

12. Applied egg-rr 28.3%

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

13. Final simplification 28.3%

    \[\leadsto \log \left({im}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im) :precision binary64 (- (/ (log (hypot re im)) (log 0.1))))

C

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

Java

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

Python

re = abs(re)
im = abs(im)
[re, im] = sort([re, im])
def code(re, im):
	return -(math.log(math.hypot(re, im)) / math.log(0.1))

Julia

re = abs(re)
im = abs(im)
re, im = sort([re, im])
function code(re, im)
	return Float64(-Float64(log(hypot(re, im)) / log(0.1)))
end

MATLAB

re = abs(re)
im = abs(im)
re, im = num2cell(sort([re, im])){:}
function tmp = code(re, im)
	tmp = -(log(hypot(re, im)) / log(0.1));
end

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
code[re_, im_] := (-N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $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. hypot-def 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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l/ 99.1%

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

   3. neg-mul-1 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

re = abs(re);
im = abs(im);
assert(re < im);
double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

re = Math.abs(re);
im = Math.abs(im);
assert re < im;
public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

re = abs(re)
im = abs(im)
[re, im] = sort([re, im])
def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

re = abs(re)
im = abs(im)
re, im = sort([re, im])
function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

re = abs(re)
im = abs(im)
re, im = num2cell(sort([re, im])){:}
function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
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. hypot-def 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. Final simplification 99.0%

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

Alternative 4: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} re = |re|\\ im = |im|\\ [re, im] = \mathsf{sort}([re, im])\\ \\ \frac{-\log im}{\log 0.1} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im) :precision binary64 (/ (- (log im)) (log 0.1)))

C

re = abs(re);
im = abs(im);
assert(re < im);
double code(double re, double im) {
	return -log(im) / log(0.1);
}

Fortran

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
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

re = Math.abs(re);
im = Math.abs(im);
assert re < im;
public static double code(double re, double im) {
	return -Math.log(im) / Math.log(0.1);
}

Python

re = abs(re)
im = abs(im)
[re, im] = sort([re, im])
def code(re, im):
	return -math.log(im) / math.log(0.1)

Julia

re = abs(re)
im = abs(im)
re, im = sort([re, im])
function code(re, im)
	return Float64(Float64(-log(im)) / log(0.1))
end

MATLAB

re = abs(re)
im = abs(im)
re, im = num2cell(sort([re, im])){:}
function tmp = code(re, im)
	tmp = -log(im) / log(0.1);
end

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
code[re_, im_] := N[((-N[Log[im], $MachinePrecision]) / N[Log[0.1], $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. hypot-def 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. Taylor expanded in re around 0 28.1%

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

   1. frac-2neg 28.1%

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

   2. div-inv 28.0%

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

   3. neg-log 28.1%

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

   4. metadata-eval 28.1%

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

6. Applied egg-rr 28.1%

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

   1. log-rec 28.1%

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

   2. associate-*r/ 28.1%

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

   3. *-rgt-identity 28.1%

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

   4. log-rec 28.1%

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

8. Simplified 28.1%

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

9. Final simplification 28.1%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} re = |re|\\ im = |im|\\ [re, im] = \mathsf{sort}([re, im])\\ \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

re = abs(re);
im = abs(im);
assert(re < im);
double code(double re, double im) {
	return log(im) / log(10.0);
}

Fortran

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
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

re = Math.abs(re);
im = Math.abs(im);
assert re < im;
public static double code(double re, double im) {
	return Math.log(im) / Math.log(10.0);
}

Python

re = abs(re)
im = abs(im)
[re, im] = sort([re, im])
def code(re, im):
	return math.log(im) / math.log(10.0)

Julia

re = abs(re)
im = abs(im)
re, im = sort([re, im])
function code(re, im)
	return Float64(log(im) / log(10.0))
end

MATLAB

re = abs(re)
im = abs(im)
re, im = num2cell(sort([re, im])){:}
function tmp = code(re, im)
	tmp = log(im) / log(10.0);
end

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
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. hypot-def 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. Taylor expanded in re around 0 28.1%

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

5. Final simplification 28.1%

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

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))