math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.5%

Time: 11.3s

Alternatives: 6

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: 51.2% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{-0.3333333333333333}{\log 0.1} \cdot 3\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* (/ -0.3333333333333333 (log 0.1)) 3.0) (log (hypot re im))))

C

double code(double re, double im) {
	return ((-0.3333333333333333 / log(0.1)) * 3.0) * log(hypot(re, im));
}

Java

public static double code(double re, double im) {
	return ((-0.3333333333333333 / Math.log(0.1)) * 3.0) * Math.log(Math.hypot(re, im));
}

Python

def code(re, im):
	return ((-0.3333333333333333 / math.log(0.1)) * 3.0) * math.log(math.hypot(re, im))

Julia

function code(re, im)
	return Float64(Float64(Float64(-0.3333333333333333 / log(0.1)) * 3.0) * log(hypot(re, im)))
end

MATLAB

function tmp = code(re, im)
	tmp = ((-0.3333333333333333 / log(0.1)) * 3.0) * log(hypot(re, im));
end

Wolfram

code[re_, im_] := N[(N[(N[(-0.3333333333333333 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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. *-un-lft-identity 99.1%

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

   2. add-sqr-sqrt 99.1%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

      \[\leadsto \frac{1}{\color{blue}{{\log 10}^{0.5}}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]

   5. pow-flip 99.2%

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

   6. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\log 10}^{-0.5} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]
6. Step-by-step derivation

   1. metadata-eval 99.2%

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

   2. pow-flip 99.2%

      \[\leadsto \color{blue}{\frac{1}{{\log 10}^{0.5}}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]

   3. pow1/2 99.2%

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

   4. times-frac 99.1%

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

   5. add-sqr-sqrt 99.1%

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

   6. associate-/l* 99.1%

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

   7. div-inv 99.1%

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

   8. associate-/r* 98.6%

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

   9. frac-2neg 98.6%

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

   10. metadata-eval 98.6%

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

   11. neg-log 98.9%

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

   12. metadata-eval 98.9%

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

7. Applied egg-rr 98.9%

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

   1. associate-/r/ 98.9%

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

   2. /-rgt-identity 98.9%

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

   3. associate-*l/ 99.0%

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

   4. metadata-eval 99.0%

      \[\leadsto \frac{\color{blue}{\left(-0.3333333333333333 \cdot 3\right)} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \]

   5. associate-*r* 99.1%

      \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{\log 0.1} \]

   6. *-commutative 99.1%

      \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)}}{\log 0.1} \]

   7. associate-*l/ 99.3%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)} \]

   8. *-commutative 99.3%

      \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]

   9. associate-*r* 99.6%

      \[\leadsto \color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot 3\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot 3\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

10. Final simplification 99.6%

    \[\leadsto \left(\frac{-0.3333333333333333}{\log 0.1} \cdot 3\right) \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 3.0 (* (/ -0.3333333333333333 (log 0.1)) (log (hypot re im)))))

C

double code(double re, double im) {
	return 3.0 * ((-0.3333333333333333 / log(0.1)) * log(hypot(re, im)));
}

Java

public static double code(double re, double im) {
	return 3.0 * ((-0.3333333333333333 / Math.log(0.1)) * Math.log(Math.hypot(re, im)));
}

Python

def code(re, im):
	return 3.0 * ((-0.3333333333333333 / math.log(0.1)) * math.log(math.hypot(re, im)))

Julia

function code(re, im)
	return Float64(3.0 * Float64(Float64(-0.3333333333333333 / log(0.1)) * log(hypot(re, im))))
end

MATLAB

function tmp = code(re, im)
	tmp = 3.0 * ((-0.3333333333333333 / log(0.1)) * log(hypot(re, im)));
end

Wolfram

code[re_, im_] := N[(3.0 * N[(N[(-0.3333333333333333 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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. add-log-exp 99.1%

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

   2. add-cube-cbrt 99.0%

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

   3. log-prod 99.0%

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

   4. pow2 99.0%

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

   5. div-inv 98.7%

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

   6. exp-to-pow 98.7%

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

   7. div-inv 98.6%

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

   8. exp-to-pow 98.5%

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

5. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]
6. Step-by-step derivation

   1. log-pow 98.5%

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

   2. distribute-lft1-in 98.5%

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

   3. metadata-eval 98.5%

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

   4. *-commutative 98.5%

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

7. Simplified 98.5%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right) \cdot 3} \]
8. Step-by-step derivation

   1. pow1/3 98.2%

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

   2. pow-pow 98.5%

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

   3. frac-2neg 98.5%

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

   4. metadata-eval 98.5%

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

   5. neg-log 99.4%

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

   6. metadata-eval 99.4%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log \color{blue}{0.1}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

9. Applied egg-rr 99.4%

   \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
10. Step-by-step derivation

    1. associate-*l/ 99.4%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\frac{-1 \cdot 0.3333333333333333}{\log 0.1}\right)}}\right) \cdot 3 \]

    2. metadata-eval 99.4%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-0.3333333333333333}}{\log 0.1}\right)}\right) \cdot 3 \]

11. Simplified 99.4%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right)} \cdot 3 \]
12. Step-by-step derivation

    1. expm1-log1p-u 72.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)\right)} \]

    2. expm1-udef 72.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)} - 1} \]

    3. log-pow 72.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \cdot 3\right)} - 1 \]

13. Applied egg-rr 72.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 72.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3\right)\right)} \]

    2. expm1-log1p 99.3%

       \[\leadsto \color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3} \]

    3. *-commutative 99.3%

       \[\leadsto \color{blue}{3 \cdot \left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]

15. Simplified 99.3%

    \[\leadsto \color{blue}{3 \cdot \left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]

16. Final simplification 99.3%

    \[\leadsto 3 \cdot \left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

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 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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 4: 27.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (/ -0.3333333333333333 (log 0.1)) (* 3.0 (log im))))

C

double code(double re, double im) {
	return (-0.3333333333333333 / log(0.1)) * (3.0 * log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.3333333333333333d0) / log(0.1d0)) * (3.0d0 * log(im))
end function

Java

public static double code(double re, double im) {
	return (-0.3333333333333333 / Math.log(0.1)) * (3.0 * Math.log(im));
}

Python

def code(re, im):
	return (-0.3333333333333333 / math.log(0.1)) * (3.0 * math.log(im))

Julia

function code(re, im)
	return Float64(Float64(-0.3333333333333333 / log(0.1)) * Float64(3.0 * log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (-0.3333333333333333 / log(0.1)) * (3.0 * log(im));
end

Wolfram

code[re_, im_] := N[(N[(-0.3333333333333333 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[(3.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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. add-log-exp 99.1%

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

   2. add-cube-cbrt 99.0%

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

   3. log-prod 99.0%

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

   4. pow2 99.0%

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

   5. div-inv 98.7%

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

   6. exp-to-pow 98.7%

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

   7. div-inv 98.6%

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

   8. exp-to-pow 98.5%

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

5. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]
6. Step-by-step derivation

   1. log-pow 98.5%

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

   2. distribute-lft1-in 98.5%

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

   3. metadata-eval 98.5%

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

   4. *-commutative 98.5%

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

7. Simplified 98.5%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right) \cdot 3} \]
8. Step-by-step derivation

   1. pow1/3 98.2%

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

   2. pow-pow 98.5%

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

   3. frac-2neg 98.5%

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

   4. metadata-eval 98.5%

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

   5. neg-log 99.4%

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

   6. metadata-eval 99.4%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log \color{blue}{0.1}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

9. Applied egg-rr 99.4%

   \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
10. Step-by-step derivation

    1. associate-*l/ 99.4%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\frac{-1 \cdot 0.3333333333333333}{\log 0.1}\right)}}\right) \cdot 3 \]

    2. metadata-eval 99.4%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-0.3333333333333333}}{\log 0.1}\right)}\right) \cdot 3 \]

11. Simplified 99.4%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right)} \cdot 3 \]
12. Step-by-step derivation

    1. expm1-log1p-u 72.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)\right)} \]

    2. expm1-udef 72.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)} - 1} \]

    3. log-pow 72.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \cdot 3\right)} - 1 \]

13. Applied egg-rr 72.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 72.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3\right)\right)} \]

    2. expm1-log1p 99.3%

       \[\leadsto \color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot 3} \]

    3. associate-*l* 99.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)} \]

15. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)} \]

16. Taylor expanded in re around 0 29.3%

    \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \color{blue}{\left(3 \cdot \log im\right)} \]
17. Step-by-step derivation

    1. *-commutative 29.3%

       \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \color{blue}{\left(\log im \cdot 3\right)} \]

18. Simplified 29.3%

    \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \color{blue}{\left(\log im \cdot 3\right)} \]

19. Final simplification 29.3%

    \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log im\right) \]

Alternative 5: 26.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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 29.3%

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

   1. clear-num 29.3%

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

   2. inv-pow 29.3%

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

6. Applied egg-rr 29.3%

   \[\leadsto \color{blue}{{\left(\frac{\log 10}{\log im}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 29.3%

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

8. Simplified 29.3%

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

9. Final simplification 29.3%

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

Alternative 6: 26.9% 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 52.1%

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

   1. +-commutative 52.1%

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

   2. +-commutative 52.1%

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

   3. sqr-neg 52.1%

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

   4. sqr-neg 52.1%

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

   5. +-commutative 52.1%

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

   6. +-commutative 52.1%

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

   7. 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 29.3%

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

5. Final simplification 29.3%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))