math.log10 on complex, real part

Percentage Accurate: 52.4% → 99.7%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. add-log-exp 99.1%

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

   2. div-inv 98.6%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 98.9%

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

   7. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. metadata-eval 98.9%

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

   2. metadata-eval 98.9%

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

   3. neg-log 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt 98.5%

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

   6. associate-/r* 99.8%

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

   7. un-div-inv 99.8%

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

   8. pow2 99.8%

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

   9. pow1/2 99.8%

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

   10. pow-flip 99.8%

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

   11. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. add-sqr-sqrt 99.1%

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

   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. Step-by-step derivation

   1. frac-times 99.1%

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

   2. add-sqr-sqrt 99.1%

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

   3. associate-/l* 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 99.1% 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 49.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 44.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-158)
   (/ 1.0 (/ (log 10.0) (log (- re))))
   (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-158) {
		tmp = 1.0 / (log(10.0) / log(-re));
	} 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 <= (-2d-158)) then
        tmp = 1.0d0 / (log(10.0d0) / log(-re))
    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 <= -2e-158) {
		tmp = 1.0 / (Math.log(10.0) / Math.log(-re));
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-158:
		tmp = 1.0 / (math.log(10.0) / math.log(-re))
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-158)
		tmp = Float64(1.0 / Float64(log(10.0) / log(Float64(-re))));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-158)
		tmp = 1.0 / (log(10.0) / log(-re));
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-158], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.00000000000000013e-158

   1. Initial program 49.9%

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

      1. hypot-def 99.2%

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

   3. Simplified 99.2%

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

      1. div-inv 98.7%

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

      2. frac-2neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. neg-log 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l/ 98.9%

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

      3. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in re around -inf 70.4%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

      3. frac-2neg 70.3%

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

      4. neg-log 70.5%

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

      5. metadata-eval 70.5%

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

      6. neg-log 70.5%

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

      7. clear-num 70.5%

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

      8. div-inv 70.5%

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

      9. metadata-eval 70.5%

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

   10. Applied egg-rr 70.5%

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

       1. unpow-1 70.5%

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

       2. *-commutative 70.5%

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

       3. mul-1-neg 70.5%

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

   12. Simplified 70.5%

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

   if -2.00000000000000013e-158 < re

   1. Initial program 49.8%

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

      1. hypot-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in re around 0 33.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5: 44.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-158) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-158) {
		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 <= (-2d-158)) 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 <= -2e-158) {
		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 <= -2e-158:
		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 <= -2e-158)
		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 <= -2e-158)
		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, -2e-158], 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 < -2.00000000000000013e-158

   1. Initial program 49.9%

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

      1. hypot-def 99.2%

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

   3. Simplified 99.2%

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

      1. div-inv 98.7%

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

      2. frac-2neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. neg-log 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l/ 98.9%

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

      3. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in re around -inf 70.4%

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

      1. frac-2neg 70.4%

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

      2. neg-log 70.5%

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

      3. metadata-eval 70.5%

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

      4. metadata-eval 70.5%

         \[\leadsto \frac{-\log \left(\frac{-1}{re}\right)}{\log \color{blue}{\left(1 + 9\right)}} \]

      5. log1p-udef 70.5%

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

      6. div-inv 70.2%

         \[\leadsto \color{blue}{\left(-\log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(9\right)}} \]

      7. neg-log 70.2%

         \[\leadsto \color{blue}{\log \left(\frac{1}{\frac{-1}{re}}\right)} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

      8. clear-num 70.2%

         \[\leadsto \log \color{blue}{\left(\frac{re}{-1}\right)} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

      9. div-inv 70.2%

         \[\leadsto \log \color{blue}{\left(re \cdot \frac{1}{-1}\right)} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

      10. metadata-eval 70.2%

          \[\leadsto \log \left(re \cdot \color{blue}{-1}\right) \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

      11. log1p-udef 70.2%

          \[\leadsto \log \left(re \cdot -1\right) \cdot \frac{1}{\color{blue}{\log \left(1 + 9\right)}} \]

      12. metadata-eval 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. associate-*r/ 70.5%

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

       2. *-rgt-identity 70.5%

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

       3. *-commutative 70.5%

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

       4. mul-1-neg 70.5%

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

   12. Simplified 70.5%

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

   if -2.00000000000000013e-158 < re

   1. Initial program 49.8%

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

      1. hypot-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in re around 0 33.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 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 49.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 28.3%

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

   1. frac-2neg 28.3%

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

   2. neg-log 28.2%

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

   3. metadata-eval 28.2%

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

   4. div-inv 28.2%

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

6. Applied egg-rr 28.2%

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

   1. log-rec 28.2%

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

   2. associate-*r/ 28.2%

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

   3. *-rgt-identity 28.2%

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

   4. log-rec 28.2%

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

8. Simplified 28.2%

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

   1. *-un-lft-identity 28.2%

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

   2. add-cube-cbrt 27.9%

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

   3. times-frac 28.0%

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

   4. pow2 28.0%

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

   5. add-sqr-sqrt 8.7%

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

   6. sqrt-unprod 9.0%

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

   7. sqr-neg 9.0%

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

   8. sqrt-unprod 0.4%

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

   9. add-sqr-sqrt 3.1%

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

10. Applied egg-rr 3.1%

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

    1. *-commutative 3.1%

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

    2. times-frac 3.1%

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

    3. *-rgt-identity 3.1%

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

    4. unpow2 3.1%

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

    5. rem-3cbrt-rft 3.1%

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

12. Simplified 3.1%

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

13. Final simplification 3.1%

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

Alternative 7: 27.5% 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 49.8%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 28.3%

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

5. Final simplification 28.3%

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

Reproduce

herbie shell --seed 2023195 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))