math.log10 on complex, real part

Percentage Accurate: 51.8% → 98.9%

Time: 7.4s

Alternatives: 4

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: 51.8% 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: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \begin{array}{l} t_0 := {\log 10}^{-0.5}\\ \left(t\_0 \cdot t\_0\right) \cdot \log im\_m \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (let* ((t_0 (pow (log 10.0) -0.5))) (* (* t_0 t_0) (log im_m))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	double t_0 = pow(log(10.0), -0.5);
	return (t_0 * t_0) * log(im_m);
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: t_0
    t_0 = log(10.0d0) ** (-0.5d0)
    code = (t_0 * t_0) * log(im_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	double t_0 = Math.pow(Math.log(10.0), -0.5);
	return (t_0 * t_0) * Math.log(im_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	t_0 = math.pow(math.log(10.0), -0.5)
	return (t_0 * t_0) * math.log(im_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	t_0 = log(10.0) ^ -0.5
	return Float64(Float64(t_0 * t_0) * log(im_m))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	t_0 = log(10.0) ^ -0.5;
	tmp = (t_0 * t_0) * log(im_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := Block[{t$95$0 = N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 26.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\log 10}\right), \color{blue}{\log im}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \log 10\right), \log \color{blue}{im}\right) \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(10\right)\right), \log im\right) \]

   6. log-lowering-log.f64 25.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(10\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

9. Applied egg-rr 25.9%

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

    1. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\log 10}^{-1}\right), \mathsf{log.f64}\left(\color{blue}{im}\right)\right) \]

    2. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\log 10}^{\left(\frac{-1}{2}\right)} \cdot {\log 10}^{\left(\frac{-1}{2}\right)}\right), \mathsf{log.f64}\left(\color{blue}{im}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\log 10}^{\left(\frac{-1}{2}\right)}\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(\color{blue}{im}\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\log 10}^{\frac{-1}{2}}\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\log 10}^{\left(\frac{1}{-2}\right)}\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\log 10, \left(\frac{1}{-2}\right)\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    7. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \left(\frac{1}{-2}\right)\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \left({\log 10}^{\left(\frac{-1}{2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \left({\log 10}^{\frac{-1}{2}}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \left({\log 10}^{\left(\frac{1}{-2}\right)}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    11. pow-lowering-pow.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\log 10, \left(\frac{1}{-2}\right)\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    12. log-lowering-log.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \left(\frac{1}{-2}\right)\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

    13. metadata-eval 26.1%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(10\right), \frac{-1}{2}\right)\right), \mathsf{log.f64}\left(im\right)\right) \]

11. Applied egg-rr 26.1%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log (hypot re_m im_m)) (log 10.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[N[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{1}{\frac{\log 10}{\log im\_m}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ 1.0 (/ (log 10.0) (log im_m))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return 1.0 / (log(10.0) / log(im_m));
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = 1.0d0 / (log(10.0d0) / log(im_m))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return 1.0 / (Math.log(10.0) / Math.log(im_m));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return 1.0 / (math.log(10.0) / math.log(im_m))

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(1.0 / Float64(log(10.0) / log(im_m)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = 1.0 / (log(10.0) / log(im_m));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 26.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\log 10}{\log im}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\log 10, \color{blue}{\log im}\right)\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(10\right), \log \color{blue}{im}\right)\right) \]

   5. log-lowering-log.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(10\right), \mathsf{log.f64}\left(im\right)\right)\right) \]

9. Applied egg-rr 26.0%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log im_m) (log 10.0)))

C

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

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m) / log(10.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 26.0%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))