math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.2%

Time: 10.5s

Alternatives: 12

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. 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 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

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. 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-cbrt-cube 98.9%

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

   2. pow3 98.9%

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

5. Applied egg-rr 98.9%

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

   1. rem-cbrt-cube 99.0%

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

   2. frac-2neg 99.0%

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

   3. neg-mul-1 99.0%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

   6. associate-/l* 99.1%

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

7. Applied egg-rr 99.1%

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

8. 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. 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 51.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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: 43.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-157)
   (/ (log (- re)) (log 10.0))
   (/ -1.0 (/ (log 0.1) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-157) {
		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 <= (-2.4d-157)) 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 <= -2.4e-157) {
		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 <= -2.4e-157:
		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 <= -2.4e-157)
		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 <= -2.4e-157)
		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, -2.4e-157], 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 < -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. 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 70.2%

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

      1. associate-*r/ 70.2%

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

      2. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

      1. add-cube-cbrt 69.5%

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

      2. *-un-lft-identity 69.5%

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

      3. times-frac 69.6%

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

   8. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 69.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      2. associate-*r/ 69.5%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10}} \]

      3. unpow2 69.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\log \left(-re\right)} \cdot \sqrt[3]{\log \left(-re\right)}\right)} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      4. rem-3cbrt-lft 70.2%

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

   10. Simplified 70.2%

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

   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. 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-cbrt-cube 98.9%

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

      2. pow3 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. rem-cbrt-cube 99.0%

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

      2. frac-2neg 99.0%

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

      3. neg-mul-1 99.0%

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

      4. neg-log 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-/l* 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. 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%

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

Alternative 6: 43.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e-157) {
		tmp = -log(-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 <= (-3.5d-157)) then
        tmp = -log(-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 <= -3.5e-157) {
		tmp = -Math.log(-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 <= -3.5e-157:
		tmp = -math.log(-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 <= -3.5e-157)
		tmp = Float64(Float64(-log(Float64(-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 <= -3.5e-157)
		tmp = -log(-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, -3.5e-157], N[((-N[Log[(-re)], $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 < -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. 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 70.2%

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

      1. associate-*r/ 70.2%

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

      2. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

      1. distribute-frac-neg 70.2%

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

      2. neg-sub0 70.2%

         \[\leadsto \color{blue}{0 - \frac{\log \left(\frac{-1}{re}\right)}{\log 10}} \]

      3. frac-2neg 70.2%

         \[\leadsto 0 - \color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{-\log 10}} \]

      4. neg-log 70.3%

         \[\leadsto 0 - \frac{-\log \left(\frac{-1}{re}\right)}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

      5. metadata-eval 70.3%

         \[\leadsto 0 - \frac{-\log \left(\frac{-1}{re}\right)}{\log \color{blue}{0.1}} \]

      6. neg-log 70.3%

         \[\leadsto 0 - \frac{\color{blue}{\log \left(\frac{1}{\frac{-1}{re}}\right)}}{\log 0.1} \]

      7. frac-2neg 70.3%

         \[\leadsto 0 - \frac{\log \left(\frac{1}{\color{blue}{\frac{--1}{-re}}}\right)}{\log 0.1} \]

      8. metadata-eval 70.3%

         \[\leadsto 0 - \frac{\log \left(\frac{1}{\frac{\color{blue}{1}}{-re}}\right)}{\log 0.1} \]

      9. remove-double-div 70.3%

         \[\leadsto 0 - \frac{\log \color{blue}{\left(-re\right)}}{\log 0.1} \]

   8. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{0 - \frac{\log \left(-re\right)}{\log 0.1}} \]
   9. Step-by-step derivation

      1. neg-sub0 70.3%

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

      2. distribute-neg-frac 70.3%

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

   10. Simplified 70.3%

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

   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. 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-cbrt-cube 98.9%

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

      2. pow3 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. rem-cbrt-cube 99.0%

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

      2. frac-2neg 99.0%

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

      3. neg-mul-1 99.0%

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

      4. neg-log 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-/l* 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. 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%

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

Alternative 7: 43.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= -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 <= (-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 <= -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 <= -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 <= -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 <= -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, -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 < -3e-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. 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 70.2%

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

      1. associate-*r/ 70.2%

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

      2. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

      1. add-cube-cbrt 69.5%

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

      2. *-un-lft-identity 69.5%

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

      3. times-frac 69.6%

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

   8. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 69.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      2. associate-*r/ 69.5%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10}} \]

      3. unpow2 69.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\log \left(-re\right)} \cdot \sqrt[3]{\log \left(-re\right)}\right)} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      4. rem-3cbrt-lft 70.2%

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

   10. Simplified 70.2%

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

   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. 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 39.1%

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

4. Final simplification 50.9%

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

Alternative 8: 43.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.2e-160) {
		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 <= (-8.2d-160)) 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 <= -8.2e-160) {
		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 <= -8.2e-160:
		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 <= -8.2e-160)
		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 <= -8.2e-160)
		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, -8.2e-160], 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 < -8.20000000000000003e-160

   1. Initial program 54.5%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in re around -inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

      1. add-cube-cbrt 69.5%

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

      2. *-un-lft-identity 69.5%

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

      3. times-frac 69.6%

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

   8. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 69.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2}} \cdot \frac{\sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      2. associate-*r/ 69.5%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\log \left(-re\right)}\right)}^{2} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10}} \]

      3. unpow2 69.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\log \left(-re\right)} \cdot \sqrt[3]{\log \left(-re\right)}\right)} \cdot \sqrt[3]{\log \left(-re\right)}}{\log 10} \]

      4. rem-3cbrt-lft 70.2%

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

   10. Simplified 70.2%

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

   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. 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 39.1%

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

      1. frac-2neg 39.1%

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

      2. div-inv 38.9%

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

      3. neg-log 39.1%

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

      4. metadata-eval 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. associate-*r/ 39.1%

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

      2. *-rgt-identity 39.1%

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

   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%

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

Alternative 9: 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. 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 31.2%

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

5. Final simplification 31.2%

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

Alternative 10: 3.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \frac{re}{\log 10 \cdot \left(im \cdot im\right)}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return 0.5 * (re * (re / (log(10.0) * (im * im))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * (re / (log(10.0d0) * (im * im))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * (re / (Math.log(10.0) * (im * im))));
}

Python

def code(re, im):
	return 0.5 * (re * (re / (math.log(10.0) * (im * im))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(re / Float64(log(10.0) * Float64(im * im)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * (re / (log(10.0) * (im * im))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(re / N[(N[Log[10.0], $MachinePrecision] * N[(im * im), $MachinePrecision]), $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. 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 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

6. Simplified 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

9. Simplified 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]

10. Taylor expanded in re around 0 2.6%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 2.6%

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

    2. unpow2 2.6%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot \left(im \cdot im\right)} \]

    3. *-commutative 2.6%

       \[\leadsto 0.5 \cdot \frac{re \cdot re}{\color{blue}{\left(im \cdot im\right) \cdot \log 10}} \]

    4. associate-*r* 2.6%

       \[\leadsto 0.5 \cdot \frac{re \cdot re}{\color{blue}{im \cdot \left(im \cdot \log 10\right)}} \]

    5. associate-*r/ 2.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \frac{re}{im \cdot \left(im \cdot \log 10\right)}\right)} \]

    6. associate-/r* 3.1%

       \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\frac{\frac{re}{im}}{im \cdot \log 10}}\right) \]

    7. *-commutative 3.1%

       \[\leadsto 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{\color{blue}{\log 10 \cdot im}}\right) \]

12. Simplified 3.1%

    \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \frac{\frac{re}{im}}{\log 10 \cdot im}\right)} \]

13. Taylor expanded in re around 0 2.9%

    \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\frac{re}{\log 10 \cdot {im}^{2}}}\right) \]
14. Step-by-step derivation

    1. *-commutative 2.9%

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

    2. unpow2 2.9%

       \[\leadsto 0.5 \cdot \left(re \cdot \frac{re}{\color{blue}{\left(im \cdot im\right)} \cdot \log 10}\right) \]

15. Simplified 2.9%

    \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\frac{re}{\left(im \cdot im\right) \cdot \log 10}}\right) \]

16. Final simplification 2.9%

    \[\leadsto 0.5 \cdot \left(re \cdot \frac{re}{\log 10 \cdot \left(im \cdot im\right)}\right) \]

Alternative 11: 3.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{\log 10 \cdot im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return 0.5 * (re * ((re / im) / (log(10.0) * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * ((re / im) / (log(10.0d0) * im)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * ((re / im) / (Math.log(10.0) * im)));
}

Python

def code(re, im):
	return 0.5 * (re * ((re / im) / (math.log(10.0) * im)))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(Float64(re / im) / Float64(log(10.0) * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * ((re / im) / (log(10.0) * im)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(re / im), $MachinePrecision] / N[(N[Log[10.0], $MachinePrecision] * im), $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. 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 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

6. Simplified 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

9. Simplified 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]

10. Taylor expanded in re around 0 2.6%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 2.6%

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

    2. unpow2 2.6%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot \left(im \cdot im\right)} \]

    3. *-commutative 2.6%

       \[\leadsto 0.5 \cdot \frac{re \cdot re}{\color{blue}{\left(im \cdot im\right) \cdot \log 10}} \]

    4. associate-*r* 2.6%

       \[\leadsto 0.5 \cdot \frac{re \cdot re}{\color{blue}{im \cdot \left(im \cdot \log 10\right)}} \]

    5. associate-*r/ 2.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \frac{re}{im \cdot \left(im \cdot \log 10\right)}\right)} \]

    6. associate-/r* 3.1%

       \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\frac{\frac{re}{im}}{im \cdot \log 10}}\right) \]

    7. *-commutative 3.1%

       \[\leadsto 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{\color{blue}{\log 10 \cdot im}}\right) \]

12. Simplified 3.1%

    \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \frac{\frac{re}{im}}{\log 10 \cdot im}\right)} \]

13. Final simplification 3.1%

    \[\leadsto 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{\log 10 \cdot im}\right) \]

Alternative 12: 3.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{re}{\log 10 \cdot im} \cdot \frac{re}{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return 0.5 * ((re / (log(10.0) * im)) * (re / im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((re / (log(10.0d0) * im)) * (re / im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((re / (Math.log(10.0) * im)) * (re / im));
}

Python

def code(re, im):
	return 0.5 * ((re / (math.log(10.0) * im)) * (re / im))

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(re / Float64(log(10.0) * im)) * Float64(re / im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((re / (log(10.0) * im)) * (re / im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(re / N[(N[Log[10.0], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * N[(re / im), $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. 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 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 26.5%

      \[\leadsto \frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

6. Simplified 26.5%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10} + 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]

7. Taylor expanded in im around 0 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{\log 10 \cdot {im}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{re \cdot re}}{\log 10 \cdot {im}^{2}} \]

   2. unpow2 2.6%

      \[\leadsto 0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \color{blue}{\left(im \cdot im\right)}} \]

9. Simplified 2.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{re \cdot re}{\log 10 \cdot \left(im \cdot im\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 2.6%

       \[\leadsto 0.5 \cdot \frac{re \cdot re}{\color{blue}{\left(\log 10 \cdot im\right) \cdot im}} \]

    2. times-frac 3.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{re}{\log 10 \cdot im} \cdot \frac{re}{im}\right)} \]

    3. *-commutative 3.2%

       \[\leadsto 0.5 \cdot \left(\frac{re}{\color{blue}{im \cdot \log 10}} \cdot \frac{re}{im}\right) \]

11. Applied egg-rr 3.2%

    \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{re}{im \cdot \log 10} \cdot \frac{re}{im}\right)} \]

12. Final simplification 3.2%

    \[\leadsto 0.5 \cdot \left(\frac{re}{\log 10 \cdot im} \cdot \frac{re}{im}\right) \]

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)))