math.log10 on complex, real part

Percentage Accurate: 51.7% → 99.5%

Time: 7.9s

Alternatives: 5

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: 51.7% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   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. expm1-log1p-u 75.1%

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

   2. expm1-udef 75.1%

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

   3. log1p-udef 75.1%

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

   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. Step-by-step derivation

   1. associate--l+ 99.1%

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

7. Simplified 99.1%

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

   1. associate-+r- 99.1%

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

   2. add-exp-log 75.1%

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

   3. log1p-udef 75.1%

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

   4. expm1-udef 75.1%

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

   5. expm1-log1p-u 99.1%

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

   6. rem-cbrt-cube 98.9%

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

   7. cube-div 98.8%

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

   8. div-inv 98.8%

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

   9. cbrt-prod 98.7%

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

   10. rem-cbrt-cube 98.6%

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

   11. pow-flip 99.6%

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

   12. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

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(1.0 + Float64(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[(1.0 + N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   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. expm1-log1p-u 75.1%

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

   2. expm1-udef 75.1%

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

   3. log1p-udef 75.1%

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

   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. Step-by-step derivation

   1. associate--l+ 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

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 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   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: 27.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   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. expm1-log1p-u 75.1%

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

   2. expm1-udef 75.1%

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

   3. log1p-udef 75.1%

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

   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. Step-by-step derivation

   1. associate--l+ 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in re around 0 31.2%

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

9. Final simplification 31.2%

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

Alternative 5: 27.6% 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 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   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 31.2%

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

5. Final simplification 31.2%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))