math.log10 on complex, real part

Percentage Accurate: 51.6% → 99.1%

Time: 10.7s

Alternatives: 9

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: 51.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\log 10}\\ \frac{1}{t_0} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (log 10.0)))) (* (/ 1.0 t_0) (/ (log (hypot re im)) t_0))))

C

double code(double re, double im) {
	double t_0 = sqrt(log(10.0));
	return (1.0 / t_0) * (log(hypot(re, im)) / t_0);
}

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(Math.log(10.0));
	return (1.0 / t_0) * (Math.log(Math.hypot(re, im)) / t_0);
}

Python

def code(re, im):
	t_0 = math.sqrt(math.log(10.0))
	return (1.0 / t_0) * (math.log(math.hypot(re, im)) / t_0)

Julia

function code(re, im)
	t_0 = sqrt(log(10.0))
	return Float64(Float64(1.0 / t_0) * Float64(log(hypot(re, im)) / t_0))
end

MATLAB

function tmp = code(re, im)
	t_0 = sqrt(log(10.0));
	tmp = (1.0 / t_0) * (log(hypot(re, im)) / t_0);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\log 0.1}{-\log \left(\mathsf{hypot}\left(re, im\right)\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (pow (/ (log 0.1) (- (log (hypot re im)))) -1.0))

C

double code(double re, double im) {
	return pow((log(0.1) / -log(hypot(re, im))), -1.0);
}

Java

public static double code(double re, double im) {
	return Math.pow((Math.log(0.1) / -Math.log(Math.hypot(re, im))), -1.0);
}

Python

def code(re, im):
	return math.pow((math.log(0.1) / -math.log(math.hypot(re, im))), -1.0)

Julia

function code(re, im)
	return Float64(log(0.1) / Float64(-log(hypot(re, im)))) ^ -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = (log(0.1) / -log(hypot(re, im))) ^ -1.0;
end

Wolfram

code[re_, im_] := N[Power[N[(N[Log[0.1], $MachinePrecision] / (-N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. clear-num 99.0%

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

   2. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

   1. frac-2neg 99.0%

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

   2. neg-log 99.1%

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

   3. metadata-eval 99.1%

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

   4. div-inv 99.0%

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

7. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

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 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l/ 99.0%

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

   3. neg-mul-1 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

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 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Final simplification 99.0%

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

Alternative 5: 43.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.6e-161)
   (/ (log (- re)) (log 10.0))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.6e-161) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = 1.0 / (log(10.0) / log(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.6d-161) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = 1.0d0 / (log(10.0d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.6e-161) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.6e-161:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.6e-161)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(1.0 / Float64(log(10.0) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.6e-161)
		tmp = log(-re) / log(10.0);
	else
		tmp = 1.0 / (log(10.0) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.6e-161], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.59999999999999993e-161

   1. Initial program 56.1%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 98.5%

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

      2. add-sqr-sqrt 66.7%

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

      3. associate-*l* 66.8%

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

      4. frac-2neg 66.8%

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

      5. metadata-eval 66.8%

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

      6. neg-log 66.9%

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

      7. metadata-eval 66.9%

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

   5. Applied egg-rr 66.9%

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

   6. Taylor expanded in re around -inf 25.9%

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

      1. frac-2neg 25.9%

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

      2. neg-log 25.9%

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

      3. metadata-eval 25.9%

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

      4. div-inv 25.8%

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

      5. neg-log 25.8%

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

      6. clear-num 25.8%

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

      7. div-inv 25.8%

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

      8. metadata-eval 25.8%

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

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\log \left(re \cdot -1\right) \cdot \frac{1}{\log 10}} \]
   9. Step-by-step derivation

      1. associate-*r/ 25.9%

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

      2. *-rgt-identity 25.9%

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

      3. *-commutative 25.9%

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

      4. mul-1-neg 25.9%

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

   10. Simplified 25.9%

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

   if 1.59999999999999993e-161 < im

   1. Initial program 57.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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. Taylor expanded in re around 0 71.5%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. unpow-1 71.3%

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

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 6: 43.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{-161}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e-161)
   (/ (log (/ -1.0 re)) (log 0.1))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e-161) {
		tmp = log((-1.0 / re)) / log(0.1);
	} else {
		tmp = 1.0 / (log(10.0) / log(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1d-161) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    else
        tmp = 1.0d0 / (log(10.0d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e-161) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e-161:
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e-161)
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(1.0 / Float64(log(10.0) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e-161)
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = 1.0 / (log(10.0) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e-161], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.00000000000000003e-161

   1. Initial program 56.1%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 98.5%

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

      2. add-sqr-sqrt 66.7%

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

      3. associate-*l* 66.8%

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

      4. frac-2neg 66.8%

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

      5. metadata-eval 66.8%

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

      6. neg-log 66.9%

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

      7. metadata-eval 66.9%

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

   5. Applied egg-rr 66.9%

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

   6. Taylor expanded in re around -inf 25.9%

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

   if 1.00000000000000003e-161 < im

   1. Initial program 57.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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. Taylor expanded in re around 0 71.5%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. unpow-1 71.3%

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

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{-161}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 7: 43.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.6e-161) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.6e-161) {
		tmp = log(-re) / log(10.0);
	} 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 (im <= 1.6d-161) then
        tmp = log(-re) / log(10.0d0)
    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 (im <= 1.6e-161) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.6e-161:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.6e-161)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.6e-161)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.6e-161], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.59999999999999993e-161

   1. Initial program 56.1%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 98.5%

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

      2. add-sqr-sqrt 66.7%

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

      3. associate-*l* 66.8%

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

      4. frac-2neg 66.8%

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

      5. metadata-eval 66.8%

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

      6. neg-log 66.9%

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

      7. metadata-eval 66.9%

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

   5. Applied egg-rr 66.9%

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

   6. Taylor expanded in re around -inf 25.9%

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

      1. frac-2neg 25.9%

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

      2. neg-log 25.9%

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

      3. metadata-eval 25.9%

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

      4. div-inv 25.8%

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

      5. neg-log 25.8%

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

      6. clear-num 25.8%

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

      7. div-inv 25.8%

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

      8. metadata-eval 25.8%

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

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\log \left(re \cdot -1\right) \cdot \frac{1}{\log 10}} \]
   9. Step-by-step derivation

      1. associate-*r/ 25.9%

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

      2. *-rgt-identity 25.9%

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

      3. *-commutative 25.9%

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

      4. mul-1-neg 25.9%

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

   10. Simplified 25.9%

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

   if 1.59999999999999993e-161 < im

   1. Initial program 57.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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. Taylor expanded in re around 0 71.5%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 8: 2.9% 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 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 68.5%

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

   3. associate-*l* 68.5%

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

   4. frac-2neg 68.5%

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

   5. metadata-eval 68.5%

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

   6. neg-log 68.8%

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

   7. metadata-eval 68.8%

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

5. Applied egg-rr 68.8%

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

6. Taylor expanded in re around 0 30.9%

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

   1. neg-mul-1 30.9%

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

   2. distribute-neg-frac 30.9%

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

8. Simplified 30.9%

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

   1. metadata-eval 30.9%

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

   2. neg-log 31.0%

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

   3. frac-2neg 31.0%

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

   4. expm1-log1p-u 19.6%

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

   5. expm1-udef 19.6%

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

   6. sub-neg 19.6%

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

   7. metadata-eval 19.6%

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

   8. +-commutative 19.6%

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

   9. log1p-udef 19.6%

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

   10. add-exp-log 31.0%

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

   11. +-commutative 31.0%

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

   12. frac-2neg 31.0%

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

   13. add-sqr-sqrt 11.2%

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

   14. sqrt-unprod 11.6%

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

   15. sqr-neg 11.6%

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

   16. sqrt-unprod 0.4%

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

   17. neg-log 0.4%

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

   18. metadata-eval 0.4%

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

   19. add-sqr-sqrt 2.3%

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

10. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{-1 + \left(\frac{\log im}{\log 0.1} + 1\right)} \]
11. Step-by-step derivation

    1. +-commutative 2.3%

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

    2. associate-+l+ 2.3%

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

    3. metadata-eval 2.3%

       \[\leadsto \frac{\log im}{\log 0.1} + \color{blue}{0} \]

    4. +-rgt-identity 2.3%

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

12. Simplified 2.3%

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

13. Final simplification 2.3%

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

Alternative 9: 26.8% 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 56.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Taylor expanded in re around 0 31.0%

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

5. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))