math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.2%

Time: 10.6s

Alternatives: 10

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

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

Java

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

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

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

Wolfram

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

Initial Program: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

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

Java

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

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.2% 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 51.4%

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

   1. *-un-lft-identity 51.4%

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

   2. add-sqr-sqrt 51.4%

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

   3. times-frac 51.5%

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

   4. hypot-def 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. expm1-log1p-u 29.0%

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

   2. hypot-def 69.8%

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

3. Applied egg-rr 69.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 99.0%

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

   2. clear-num 99.0%

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

   3. frac-2neg 99.0%

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

   4. metadata-eval 99.0%

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

   5. distribute-neg-frac 99.0%

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

   6. neg-log 99.1%

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

   7. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

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 51.4%

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

   1. frac-2neg 51.4%

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

   2. div-inv 51.2%

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

   3. hypot-def 98.5%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

3. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

5. Simplified 99.0%

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

6. 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 51.4%

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

   1. add-log-exp 6.3%

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

   2. *-un-lft-identity 6.3%

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

   3. log-prod 6.3%

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

   4. metadata-eval 6.3%

      \[\leadsto \frac{\log \left(\color{blue}{0} + \log \left(e^{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10} \]

   5. add-log-exp 51.4%

      \[\leadsto \frac{\log \left(0 + \color{blue}{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   6. hypot-def 99.0%

      \[\leadsto \frac{\log \left(0 + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)}{\log 10} \]

3. Applied egg-rr 99.0%

   \[\leadsto \frac{\log \color{blue}{\left(0 + \mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
4. Step-by-step derivation

   1. +-lft-identity 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \frac{\log im \cdot -0.3333333333333333}{\log 0.1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5e-157)
   (/ (log (/ -1.0 re)) (log 0.1))
   (* 3.0 (/ (* (log im) -0.3333333333333333) (log 0.1)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e-157) {
		tmp = log((-1.0 / re)) / log(0.1);
	} else {
		tmp = 3.0 * ((log(im) * -0.3333333333333333) / log(0.1));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.5d-157)) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    else
        tmp = 3.0d0 * ((log(im) * (-0.3333333333333333d0)) / log(0.1d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.5e-157) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = 3.0 * ((Math.log(im) * -0.3333333333333333) / Math.log(0.1));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.5e-157:
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = 3.0 * ((math.log(im) * -0.3333333333333333) / math.log(0.1))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5e-157)
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(3.0 * Float64(Float64(log(im) * -0.3333333333333333) / log(0.1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.5e-157)
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = 3.0 * ((log(im) * -0.3333333333333333) / log(0.1));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5e-157], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(N[Log[im], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5000000000000002e-157

   1. Initial program 54.5%

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

      1. frac-2neg 54.5%

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

      2. div-inv 54.3%

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

      3. hypot-def 98.5%

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

      4. neg-log 99.1%

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

      5. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}} \]
   4. 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%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in re around -inf 70.3%

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

   if -3.5000000000000002e-157 < re

   1. Initial program 49.6%

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

      1. add-log-exp 49.6%

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

      2. add-cube-cbrt 49.6%

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

      3. log-prod 49.5%

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

   3. Applied egg-rr 98.5%

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

      1. log-prod 98.5%

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

      2. count-2 98.5%

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

      3. distribute-lft1-in 98.5%

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

      4. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in re around 0 39.0%

      \[\leadsto \color{blue}{3 \cdot \log \left({\left(e^{\frac{\log im}{\log 10}}\right)}^{0.3333333333333333}\right)} \]
   7. Step-by-step derivation

      1. log-pow 38.9%

         \[\leadsto 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\frac{\log im}{\log 10}}\right)\right)} \]

      2. rem-log-exp 39.0%

         \[\leadsto 3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{\log im}{\log 10}}\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{3 \cdot \left(0.3333333333333333 \cdot \frac{\log im}{\log 10}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 39.0%

         \[\leadsto 3 \cdot \color{blue}{\frac{0.3333333333333333 \cdot \log im}{\log 10}} \]

      2. frac-2neg 39.0%

         \[\leadsto 3 \cdot \color{blue}{\frac{-0.3333333333333333 \cdot \log im}{-\log 10}} \]

      3. neg-log 39.2%

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

      4. metadata-eval 39.2%

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

   10. Applied egg-rr 39.2%

       \[\leadsto 3 \cdot \color{blue}{\frac{-0.3333333333333333 \cdot \log im}{\log 0.1}} \]
   11. Step-by-step derivation

       1. distribute-rgt-neg-in 39.2%

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

       2. log-rec 39.2%

          \[\leadsto 3 \cdot \frac{0.3333333333333333 \cdot \color{blue}{\log \left(\frac{1}{im}\right)}}{\log 0.1} \]

       3. *-commutative 39.2%

          \[\leadsto 3 \cdot \frac{\color{blue}{\log \left(\frac{1}{im}\right) \cdot 0.3333333333333333}}{\log 0.1} \]

       4. log-rec 39.2%

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

       5. distribute-lft-neg-out 39.2%

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

       6. distribute-rgt-neg-in 39.2%

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

       7. metadata-eval 39.2%

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

   12. Simplified 39.2%

       \[\leadsto 3 \cdot \color{blue}{\frac{\log im \cdot -0.3333333333333333}{\log 0.1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \frac{\log im \cdot -0.3333333333333333}{\log 0.1}\\ \end{array} \]

Alternative 6: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.2e-160)
   (/ (log (- re)) (log 10.0))
   (/ -1.0 (/ (log 0.1) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.2e-160) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = -1.0 / (log(0.1) / 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 (re <= (-8.2d-160)) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = (-1.0d0) / (log(0.1d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.2e-160) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = -1.0 / (Math.log(0.1) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.2e-160:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = -1.0 / (math.log(0.1) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.2e-160)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(-1.0 / Float64(log(0.1) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.2e-160)
		tmp = log(-re) / log(10.0);
	else
		tmp = -1.0 / (log(0.1) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.2e-160], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.20000000000000003e-160

   1. Initial program 54.5%

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

   2. Taylor expanded in re around -inf 70.2%

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

      1. mul-1-neg 70.2%

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

   4. Simplified 70.2%

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

   if -8.20000000000000003e-160 < re

   1. Initial program 49.6%

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

      1. expm1-log1p-u 24.1%

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

      2. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.0%

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

      2. clear-num 99.0%

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

      3. frac-2neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. neg-log 99.1%

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

      7. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in re around 0 39.1%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\log 0.1}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 7: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-157)
   (/ (log (/ -1.0 re)) (log 0.1))
   (/ -1.0 (/ (log 0.1) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-157) {
		tmp = log((-1.0 / re)) / log(0.1);
	} else {
		tmp = -1.0 / (log(0.1) / 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 (re <= (-2.4d-157)) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    else
        tmp = (-1.0d0) / (log(0.1d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-157) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = -1.0 / (Math.log(0.1) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e-157:
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = -1.0 / (math.log(0.1) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-157)
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(-1.0 / Float64(log(0.1) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-157)
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = -1.0 / (log(0.1) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-157], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.4e-157

   1. Initial program 54.5%

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

      1. frac-2neg 54.5%

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

      2. div-inv 54.3%

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

      3. hypot-def 98.5%

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

      4. neg-log 99.1%

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

      5. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}} \]
   4. 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%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in re around -inf 70.3%

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

   if -2.4e-157 < re

   1. Initial program 49.6%

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

      1. expm1-log1p-u 24.1%

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

      2. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.0%

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

      2. clear-num 99.0%

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

      3. frac-2neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. neg-log 99.1%

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

      7. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in re around 0 39.1%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\log 0.1}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 8: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-157) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-157) {
		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 (re <= (-3.3d-157)) 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 (re <= -3.3e-157) {
		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 re <= -3.3e-157:
		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 (re <= -3.3e-157)
		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 (re <= -3.3e-157)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-157], 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 re < -3.29999999999999999e-157

   1. Initial program 54.5%

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

   2. Taylor expanded in re around -inf 70.2%

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

      1. mul-1-neg 70.2%

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

   4. Simplified 70.2%

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

   if -3.29999999999999999e-157 < re

   1. Initial program 49.6%

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

   2. Taylor expanded in re around 0 39.1%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 9: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e-157) (/ (log (- re)) (log 10.0)) (/ (- (log im)) (log 0.1))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e-157) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = -log(im) / log(0.1);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3d-157)) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = -log(im) / log(0.1d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3e-157) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = -Math.log(im) / Math.log(0.1);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e-157:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = -math.log(im) / math.log(0.1)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e-157)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(Float64(-log(im)) / log(0.1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e-157)
		tmp = log(-re) / log(10.0);
	else
		tmp = -log(im) / log(0.1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e-157], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[((-N[Log[im], $MachinePrecision]) / N[Log[0.1], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3e-157

   1. Initial program 54.5%

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

   2. Taylor expanded in re around -inf 70.2%

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

      1. mul-1-neg 70.2%

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

   4. Simplified 70.2%

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

   if -3e-157 < re

   1. Initial program 49.6%

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

      1. frac-2neg 49.6%

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

      2. div-inv 49.4%

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

      3. hypot-def 98.5%

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

      4. neg-log 98.9%

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

      5. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in re around 0 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-neg-frac 39.1%

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

   8. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{-\log im}{\log 0.1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 10: 27.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

2. Taylor expanded in re around 0 31.2%

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

3. Final simplification 31.2%

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

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))