math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.1%

Time: 8.7s

Alternatives: 5

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 51.4% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right) + -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1.0 + N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 46.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. expm1-log1p-u 78.8%

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

   2. expm1-udef 78.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)} - 1} \]

   3. log1p-udef 78.7%

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

   4. add-exp-log 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 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 46.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 43.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e-10) (/ (log (/ -1.0 re)) (log 0.1)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-10) {
		tmp = log((-1.0 / re)) / log(0.1);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.8d-10)) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-10) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8e-10:
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e-10)
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.8e-10)
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e-10], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.8000000000000003e-10

   1. Initial program 41.2%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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. add-sqr-sqrt 97.1%

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

      3. associate-*l* 97.1%

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

      4. frac-2neg 97.1%

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

      5. metadata-eval 97.1%

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

      6. neg-log 97.5%

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

      7. metadata-eval 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in re around -inf 79.7%

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

   if -6.8000000000000003e-10 < re

   1. Initial program 49.0%

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

      1. hypot-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in re around 0 30.9%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 4: 3.1% 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) / 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 46.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. div-inv 98.6%

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

   2. add-sqr-sqrt 78.5%

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

   3. associate-*l* 78.5%

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

   4. frac-2neg 78.5%

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

   5. metadata-eval 78.5%

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

   6. neg-log 78.8%

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

   7. metadata-eval 78.8%

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

5. Applied egg-rr 78.8%

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

6. Taylor expanded in re around 0 25.6%

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

   1. neg-mul-1 25.6%

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

   2. distribute-neg-frac 25.6%

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

8. Simplified 25.6%

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

   1. metadata-eval 25.6%

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

   2. neg-log 25.6%

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

   3. frac-2neg 25.6%

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

   4. expm1-log1p-u 17.2%

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

   5. expm1-udef 17.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log im}{\log 10}\right)} - 1} \]

   6. frac-2neg 17.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\log im}{-\log 10}}\right)} - 1 \]

   7. add-sqr-sqrt 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\log im} \cdot \sqrt{-\log im}}}{-\log 10}\right)} - 1 \]

   8. sqrt-unprod 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\log im\right) \cdot \left(-\log im\right)}}}{-\log 10}\right)} - 1 \]

   9. sqr-neg 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\log im \cdot \log im}}}{-\log 10}\right)} - 1 \]

   10. sqrt-unprod 0.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\log im} \cdot \sqrt{\log im}}}{-\log 10}\right)} - 1 \]

   11. add-sqr-sqrt 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\log im}}{-\log 10}\right)} - 1 \]

   12. log1p-expm1-u 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\log 10\right)\right)}}\right)} - 1 \]

   13. neg-log 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log \left(\frac{1}{10}\right)}\right)\right)}\right)} - 1 \]

   14. metadata-eval 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{0.1}\right)\right)}\right)} - 1 \]

   15. expm1-udef 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(\color{blue}{e^{\log 0.1} - 1}\right)}\right)} - 1 \]

   16. add-exp-log 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(\color{blue}{0.1} - 1\right)}\right)} - 1 \]

   17. metadata-eval 2.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(\color{blue}{-0.9}\right)}\right)} - 1 \]

10. Applied egg-rr 2.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 2.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}\right)\right)} \]

    2. expm1-log1p 3.0%

       \[\leadsto \color{blue}{\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}} \]

12. Simplified 3.0%

    \[\leadsto \color{blue}{\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}} \]

13. Taylor expanded in im around 0 3.0%

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

14. Final simplification 3.0%

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

Alternative 5: 27.3% 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 46.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 25.6%

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

5. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))