math.log10 on complex, real part

Percentage Accurate: 52.5% → 99.7%

Time: 11.2s

Alternatives: 11

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: 52.5% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (pow (pow (log 10.0) -0.5) 2.0))))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), pow(pow(log(10.0), -0.5), 2.0)));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(Math.hypot(re, im), Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0)));
}

Python

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), math.pow(math.pow(math.log(10.0), -0.5), 2.0)))

Julia

function code(re, im)
	return log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = log((hypot(re, im) ^ ((log(10.0) ^ -0.5) ^ 2.0)));
end

Wolfram

code[re_, im_] := N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.9%

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

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

   2. div-inv 98.5%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. metadata-eval 99.0%

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

   2. metadata-eval 99.0%

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

   3. neg-log 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt 98.5%

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

   6. metadata-eval 98.5%

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

   7. frac-times 99.7%

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

   8. pow2 99.7%

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

   9. pow1/2 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left(\frac{1}{\color{blue}{{\log 10}^{0.5}}}\right)}^{2}\right)}\right) \]

   10. pow-flip 99.7%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\color{blue}{\left({\log 10}^{\left(-0.5\right)}\right)}}^{2}\right)}\right) \]

   11. metadata-eval 99.7%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{\color{blue}{-0.5}}\right)}^{2}\right)}\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]

8. Final simplification 99.7%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. frac-times 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. associate-/l* 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

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 53.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 53.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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: 44.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{\frac{-\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.3e-29)
   (/ 1.0 (/ (- (log 10.0)) (log (/ -1.0 re))))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.3e-29) {
		tmp = 1.0 / (-log(10.0) / log((-1.0 / re)));
	} 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 <= 4.3d-29) then
        tmp = 1.0d0 / (-log(10.0d0) / log(((-1.0d0) / re)))
    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 <= 4.3e-29) {
		tmp = 1.0 / (-Math.log(10.0) / Math.log((-1.0 / re)));
	} else {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.3e-29:
		tmp = 1.0 / (-math.log(10.0) / math.log((-1.0 / re)))
	else:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.3e-29)
		tmp = Float64(1.0 / Float64(Float64(-log(10.0)) / log(Float64(-1.0 / re))));
	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 <= 4.3e-29)
		tmp = 1.0 / (-log(10.0) / log((-1.0 / re)));
	else
		tmp = 1.0 / (log(10.0) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.3e-29], N[(1.0 / N[((-N[Log[10.0], $MachinePrecision]) / N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $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 < 4.2999999999999998e-29

   1. Initial program 59.2%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 99.0%

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

      2. div-inv 98.5%

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

      3. exp-to-pow 98.5%

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

      4. frac-2neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. neg-log 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. metadata-eval 99.0%

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

      2. metadata-eval 99.0%

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

      3. neg-log 98.5%

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

      4. frac-2neg 98.5%

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

      5. add-sqr-sqrt 98.5%

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

      6. metadata-eval 98.5%

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

      7. frac-times 99.7%

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

      8. pow2 99.7%

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

      9. pow1/2 99.7%

         \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left(\frac{1}{\color{blue}{{\log 10}^{0.5}}}\right)}^{2}\right)}\right) \]

      10. pow-flip 99.7%

          \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\color{blue}{\left({\log 10}^{\left(-0.5\right)}\right)}}^{2}\right)}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left({\left({\log 10}^{\color{blue}{-0.5}}\right)}^{2}\right)}\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. log-pow 99.6%

         \[\leadsto \color{blue}{{\left({\log 10}^{-0.5}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

      2. pow-pow 98.5%

         \[\leadsto \color{blue}{{\log 10}^{\left(-0.5 \cdot 2\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

      3. metadata-eval 98.5%

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

      4. inv-pow 98.5%

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

      5. associate-/r/ 99.0%

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

      6. log1p-expm1-u 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

      7. expm1-udef 99.0%

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

      8. add-exp-log 99.0%

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

      9. metadata-eval 99.0%

         \[\leadsto \frac{1}{\frac{\mathsf{log1p}\left(\color{blue}{9}\right)}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

   10. Taylor expanded in re around -inf 35.0%

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

       1. associate-*r/ 35.0%

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

       2. neg-mul-1 35.0%

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

   12. Simplified 35.0%

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

   if 4.2999999999999998e-29 < im

   1. Initial program 42.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 77.1%

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

      1. clear-num 77.2%

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

      2. inv-pow 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{\frac{-\log 10}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 6: 44.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4e-29)
   (/ (- (log (/ -1.0 re))) (log 10.0))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4e-29) {
		tmp = -log((-1.0 / 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 <= 4.4d-29) then
        tmp = -log(((-1.0d0) / 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 <= 4.4e-29) {
		tmp = -Math.log((-1.0 / 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 <= 4.4e-29:
		tmp = -math.log((-1.0 / 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 <= 4.4e-29)
		tmp = Float64(Float64(-log(Float64(-1.0 / 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 <= 4.4e-29)
		tmp = -log((-1.0 / 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, 4.4e-29], N[((-N[Log[N[(-1.0 / re), $MachinePrecision]], $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 < 4.39999999999999981e-29

   1. Initial program 59.2%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in re around -inf 35.0%

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

      1. associate-*r/ 35.0%

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

      2. mul-1-neg 35.0%

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

   6. Simplified 35.0%

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

   if 4.39999999999999981e-29 < im

   1. Initial program 42.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 77.1%

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

      1. clear-num 77.2%

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

      2. inv-pow 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 7: 44.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4e-29)
   (/ (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 <= 4.4e-29) {
		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 <= 4.4d-29) 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 <= 4.4e-29) {
		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 <= 4.4e-29:
		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 <= 4.4e-29)
		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 <= 4.4e-29)
		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, 4.4e-29], 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 < 4.39999999999999981e-29

   1. Initial program 59.2%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 99.0%

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

      2. div-inv 98.5%

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

      3. exp-to-pow 98.5%

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

      4. frac-2neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. neg-log 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in re around -inf 35.0%

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

   if 4.39999999999999981e-29 < im

   1. Initial program 42.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 77.1%

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

      1. clear-num 77.2%

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

      2. inv-pow 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 8: 27.6% 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 53.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 28.8%

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

   1. clear-num 28.8%

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

   2. inv-pow 28.8%

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

6. Applied egg-rr 28.8%

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

   1. unpow-1 28.8%

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

8. Simplified 28.8%

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

9. Final simplification 28.8%

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

Alternative 9: 27.6% 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 53.9%

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

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

   2. div-inv 98.5%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

6. Taylor expanded in re around 0 28.8%

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

   1. neg-mul-1 28.8%

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

   2. distribute-neg-frac 28.8%

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

8. Simplified 28.8%

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

9. Final simplification 28.8%

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

Alternative 10: 3.0% 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(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 53.9%

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

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

   2. div-inv 98.5%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

6. Taylor expanded in re around 0 28.8%

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

   1. neg-mul-1 28.8%

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

   2. distribute-neg-frac 28.8%

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

8. Simplified 28.8%

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

   1. add-sqr-sqrt 5.9%

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

   2. sqrt-unprod 6.4%

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

   3. sqr-neg 6.4%

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

   4. sqrt-unprod 0.5%

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

   5. add-sqr-sqrt 3.4%

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

   6. add-cube-cbrt 3.4%

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

   7. *-un-lft-identity 3.4%

      \[\leadsto \frac{\left(\sqrt[3]{\log im} \cdot \sqrt[3]{\log im}\right) \cdot \sqrt[3]{\log im}}{\color{blue}{1 \cdot \log 0.1}} \]

   8. times-frac 3.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\log im} \cdot \sqrt[3]{\log im}}{1} \cdot \frac{\sqrt[3]{\log im}}{\log 0.1}} \]

   9. pow2 3.4%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\log im}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\log im}}{\log 0.1} \]

10. Applied egg-rr 3.4%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log im}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\log im}}{\log 0.1}} \]
11. Step-by-step derivation

    1. /-rgt-identity 3.4%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\log im}\right)}^{2}} \cdot \frac{\sqrt[3]{\log im}}{\log 0.1} \]

    2. associate-*r/ 3.4%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log im}\right)}^{2} \cdot \sqrt[3]{\log im}}{\log 0.1}} \]

    3. unpow2 3.4%

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

    4. rem-3cbrt-lft 3.4%

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

12. Simplified 3.4%

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

13. Final simplification 3.4%

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

Alternative 11: 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 53.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 28.8%

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

5. Final simplification 28.8%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))