math.log10 on complex, real part

Percentage Accurate: 51.3% → 99.6%

Time: 11.8s

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: 51.3% 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: 99.6% 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 57.5%

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

   1. frac-2neg 99.0%

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

   2. div-inv 98.5%

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

   3. neg-log 99.0%

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

   4. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

8. Simplified 99.1%

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

   1. metadata-eval 99.1%

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

   2. neg-log 99.0%

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

   3. frac-2neg 99.0%

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

   4. rem-cbrt-cube 99.0%

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

   5. rem-cbrt-cube 97.9%

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

   6. cbrt-div 99.0%

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

   7. pow1/3 68.8%

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

   8. div-inv 68.8%

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

   9. unpow-prod-down 68.9%

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

   10. pow1/3 99.1%

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

   11. rem-cbrt-cube 99.6%

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

   12. pow-flip 99.6%

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

   13. metadata-eval 99.6%

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

10. Applied egg-rr 99.6%

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

    1. unpow1/3 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 27.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\log 10}^{-2}} \cdot \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sqrt (pow (log 10.0) -2.0)) (log im)))

C

double code(double re, double im) {
	return sqrt(pow(log(10.0), -2.0)) * log(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((log(10.0d0) ** (-2.0d0))) * log(im)
end function

Java

public static double code(double re, double im) {
	return Math.sqrt(Math.pow(Math.log(10.0), -2.0)) * Math.log(im);
}

Python

def code(re, im):
	return math.sqrt(math.pow(math.log(10.0), -2.0)) * math.log(im)

Julia

function code(re, im)
	return Float64(sqrt((log(10.0) ^ -2.0)) * log(im))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((log(10.0) ^ -2.0)) * log(im);
end

Wolfram

code[re_, im_] := N[(N[Sqrt[N[Power[N[Log[10.0], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Add Preprocessing

5. Taylor expanded in re around 0 27.8%

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

   1. clear-num 27.8%

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

   2. associate-/r/ 27.7%

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

7. Applied egg-rr 27.7%

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

   1. add-sqr-sqrt 28.0%

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

   2. sqrt-unprod 27.7%

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

   3. inv-pow 27.7%

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

   4. inv-pow 27.7%

      \[\leadsto \sqrt{{\log 10}^{-1} \cdot \color{blue}{{\log 10}^{-1}}} \cdot \log im \]

   5. pow-prod-up 28.0%

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

   6. metadata-eval 28.0%

      \[\leadsto \sqrt{{\log 10}^{\color{blue}{-2}}} \cdot \log im \]

9. Applied egg-rr 28.0%

   \[\leadsto \color{blue}{\sqrt{{\log 10}^{-2}}} \cdot \log im \]

10. Final simplification 28.0%

    \[\leadsto \sqrt{{\log 10}^{-2}} \cdot \log im \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. frac-2neg 99.0%

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

   2. div-inv 98.5%

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

   3. neg-log 99.0%

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

   4. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 4: 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 57.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Add Preprocessing

5. Final simplification 99.0%

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

Alternative 5: 27.0% 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(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 57.5%

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

   1. frac-2neg 99.0%

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

   2. div-inv 98.5%

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

   3. neg-log 99.0%

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

   4. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in re around 0 27.8%

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

    1. neg-mul-1 27.8%

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

    2. distribute-neg-frac 27.8%

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

11. Simplified 27.8%

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

12. Final simplification 27.8%

    \[\leadsto \frac{-\log im}{\log 0.1} \]
13. Add Preprocessing

Alternative 6: 27.0% 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 57.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Add Preprocessing

5. Taylor expanded in re around 0 27.8%

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

6. Final simplification 27.8%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))