math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.0%

Time: 8.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.4% 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: 99.0% 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 53.0%

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

   1. hypot-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

   2. clear-num 99.0%

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

   3. frac-2neg 99.0%

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

   4. neg-mul-1 99.0%

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

   5. neg-log 99.0%

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

   6. metadata-eval 99.0%

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

   7. 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 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 4: 43.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.2e-99)
   (/ (log (- re)) (log 10.0))
   (/ -1.0 (/ (log 0.1) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.2e-99) {
		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 (im <= 7.2d-99) 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 (im <= 7.2e-99) {
		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 im <= 7.2e-99:
		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 (im <= 7.2e-99)
		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 (im <= 7.2e-99)
		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[im, 7.2e-99], 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 im < 7.2000000000000001e-99

   1. Initial program 52.4%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 -inf 31.4%

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

      1. associate-*r/ 31.4%

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

      2. mul-1-neg 31.4%

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

   6. Simplified 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.5%

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

      4. neg-log 31.5%

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

      5. frac-2neg 31.5%

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

      6. metadata-eval 31.5%

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

      7. remove-double-div 31.5%

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

   8. Applied egg-rr 31.5%

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

      1. times-frac 31.4%

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

      2. *-lft-identity 31.4%

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

      3. rem-square-sqrt 31.4%

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

   10. Simplified 31.4%

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

   if 7.2000000000000001e-99 < im

   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. Step-by-step derivation

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow-1 98.9%

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

      2. clear-num 99.0%

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

      3. frac-2neg 99.0%

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

      4. neg-mul-1 99.0%

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

      5. neg-log 99.1%

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

      6. metadata-eval 99.1%

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

      7. associate-/l* 99.2%

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in re around 0 75.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\log 0.1}{\log im}}\\ \end{array} \]

Alternative 5: 43.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.2e-99) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.2e-99) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.2d-99) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.2e-99) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.2e-99:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.2e-99)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.2e-99)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.2e-99], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.2000000000000001e-99

   1. Initial program 52.4%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 -inf 31.4%

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

      1. associate-*r/ 31.4%

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

      2. mul-1-neg 31.4%

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

   6. Simplified 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.5%

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

      4. neg-log 31.5%

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

      5. frac-2neg 31.5%

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

      6. metadata-eval 31.5%

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

      7. remove-double-div 31.5%

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

   8. Applied egg-rr 31.5%

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

      1. times-frac 31.4%

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

      2. *-lft-identity 31.4%

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

      3. rem-square-sqrt 31.4%

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

   10. Simplified 31.4%

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

   if 7.2000000000000001e-99 < im

   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 0 75.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 43.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.2e-99) (/ (log (- re)) (log 10.0)) (/ (- (log im)) (log 0.1))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.2e-99) {
		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 (im <= 7.2d-99) 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 (im <= 7.2e-99) {
		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 im <= 7.2e-99:
		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 (im <= 7.2e-99)
		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 (im <= 7.2e-99)
		tmp = log(-re) / log(10.0);
	else
		tmp = -log(im) / log(0.1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.2e-99], 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 im < 7.2000000000000001e-99

   1. Initial program 52.4%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 -inf 31.4%

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

      1. associate-*r/ 31.4%

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

      2. mul-1-neg 31.4%

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

   6. Simplified 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.5%

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

      4. neg-log 31.5%

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

      5. frac-2neg 31.5%

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

      6. metadata-eval 31.5%

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

      7. remove-double-div 31.5%

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

   8. Applied egg-rr 31.5%

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

      1. times-frac 31.4%

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

      2. *-lft-identity 31.4%

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

      3. rem-square-sqrt 31.4%

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

   10. Simplified 31.4%

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

   if 7.2000000000000001e-99 < im

   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. Step-by-step derivation

      1. div-inv 98.4%

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

      2. frac-2neg 98.4%

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

      3. metadata-eval 98.4%

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

      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/ 99.1%

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

      3. neg-mul-1 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in re around 0 75.6%

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

      1. neg-mul-1 75.6%

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

      2. distribute-neg-frac 75.6%

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

   10. Simplified 75.6%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{\log 0.1}\\ \end{array} \]

Alternative 7: 3.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 0.1} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(0.1d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. Taylor expanded in re around 0 24.1%

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

   1. neg-mul-1 24.1%

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

   2. distribute-neg-frac 24.1%

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

10. Simplified 24.1%

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

    1. expm1-log1p-u 17.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log im}{\log 0.1}\right)\right)} \]

    2. expm1-udef 17.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\log im}{\log 0.1}\right)} - 1} \]

12. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 2.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}\right)\right)} \]

    2. expm1-log1p 2.9%

       \[\leadsto \color{blue}{\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}} \]

14. Simplified 2.9%

    \[\leadsto \color{blue}{\frac{\log im}{\mathsf{log1p}\left(-0.9\right)}} \]

15. Taylor expanded in im around 0 2.9%

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

16. Final simplification 2.9%

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

Alternative 8: 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 53.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 24.2%

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

5. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2023193 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))