math.log10 on complex, real part

Percentage Accurate: 51.1% → 99.7%

Time: 9.1s

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: 51.1% 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 52.0%

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

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

   2. div-inv 98.6%

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

   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. metadata-eval 98.5%

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

   6. add-sqr-sqrt 98.5%

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

   7. frac-times 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. inv-pow 99.8%

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

   10. sqrt-pow2 99.8%

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

   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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. expm1-log1p-u 75.2%

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

   2. expm1-udef 75.2%

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

   3. log1p-udef 75.2%

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

   4. add-exp-log 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

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 52.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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: 42.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{-90} \lor \neg \left(im \leq 5.3 \cdot 10^{-48}\right) \land im \leq 0.072:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 6e-90) (and (not (<= im 5.3e-48)) (<= im 0.072)))
   (/ 1.0 (/ (log1p 9.0) (log (- re))))
   (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 6e-90) || (!(im <= 5.3e-48) && (im <= 0.072))) {
		tmp = 1.0 / (log1p(9.0) / log(-re));
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 6e-90) || (!(im <= 5.3e-48) && (im <= 0.072))) {
		tmp = 1.0 / (Math.log1p(9.0) / Math.log(-re));
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 6e-90) or (not (im <= 5.3e-48) and (im <= 0.072)):
		tmp = 1.0 / (math.log1p(9.0) / math.log(-re))
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 6e-90) || (!(im <= 5.3e-48) && (im <= 0.072)))
		tmp = Float64(1.0 / Float64(log1p(9.0) / log(Float64(-re))));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 6e-90], And[N[Not[LessEqual[im, 5.3e-48]], $MachinePrecision], LessEqual[im, 0.072]]], N[(1.0 / N[(N[Log[1 + 9.0], $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 im < 6.00000000000000041e-90 or 5.3e-48 < im < 0.0719999999999999946

   1. Initial program 57.5%

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

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

      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. Taylor expanded in re around -inf 33.6%

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

      1. associate-*r/ 33.6%

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

      2. mul-1-neg 33.6%

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

   8. Simplified 33.6%

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

      1. rem-cbrt-cube 33.7%

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

      2. add-sqr-sqrt 24.3%

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

      3. sqrt-unprod 25.9%

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

      4. sqr-neg 25.9%

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

      5. sqrt-unprod 1.5%

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

      6. add-sqr-sqrt 1.9%

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

      7. clear-num 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. log1p-udef 1.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{log1p}\left(3 \cdot 3\right)}}{\log \left(\frac{-1}{re}\right)}} \]

      11. metadata-eval 1.9%

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

      12. add-sqr-sqrt 1.5%

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

      13. sqrt-unprod 25.9%

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

      14. sqr-neg 25.9%

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

      15. sqrt-unprod 24.3%

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

      16. add-sqr-sqrt 33.7%

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

      17. neg-log 33.7%

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

      18. frac-2neg 33.7%

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

      19. metadata-eval 33.7%

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

      20. remove-double-div 33.7%

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

   10. Applied egg-rr 33.7%

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

   if 6.00000000000000041e-90 < im < 5.3e-48 or 0.0719999999999999946 < im

   1. Initial program 37.8%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 79.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{-90} \lor \neg \left(im \leq 5.3 \cdot 10^{-48}\right) \land im \leq 0.072:\\ \;\;\;\;\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 5: 42.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.08 \cdot 10^{-90} \lor \neg \left(im \leq 2.6 \cdot 10^{-33}\right) \land im \leq 0.112:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1.08e-90) (and (not (<= im 2.6e-33)) (<= im 0.112)))
   (/ (log (/ -1.0 re)) (log 0.1))
   (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1.08e-90) || (!(im <= 2.6e-33) && (im <= 0.112))) {
		tmp = log((-1.0 / re)) / log(0.1);
	} 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 <= 1.08d-90) .or. (.not. (im <= 2.6d-33)) .and. (im <= 0.112d0)) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    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 <= 1.08e-90) || (!(im <= 2.6e-33) && (im <= 0.112))) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 1.08e-90) or (not (im <= 2.6e-33) and (im <= 0.112)):
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1.08e-90) || (!(im <= 2.6e-33) && (im <= 0.112)))
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 1.08e-90) || (~((im <= 2.6e-33)) && (im <= 0.112)))
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1.08e-90], And[N[Not[LessEqual[im, 2.6e-33]], $MachinePrecision], LessEqual[im, 0.112]]], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.0799999999999999e-90 or 2.59999999999999994e-33 < im < 0.112000000000000002

   1. Initial program 57.3%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

      1. div-inv 98.6%

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

      2. frac-2neg 98.6%

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

      3. metadata-eval 98.6%

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

      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/ 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 -inf 33.8%

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

   if 1.0799999999999999e-90 < im < 2.59999999999999994e-33 or 0.112000000000000002 < im

   1. Initial program 38.6%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   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 78.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.08 \cdot 10^{-90} \lor \neg \left(im \leq 2.6 \cdot 10^{-33}\right) \land im \leq 0.112:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 2.9% 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 52.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 25.4%

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

   1. frac-2neg 25.4%

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

   2. neg-log 25.3%

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

   3. metadata-eval 25.3%

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

   4. div-inv 25.3%

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

6. Applied egg-rr 25.3%

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

   1. log-rec 25.3%

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

   2. associate-*r/ 25.3%

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

   3. *-rgt-identity 25.3%

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

   4. log-rec 25.3%

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

8. Simplified 25.3%

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

   1. add-sqr-sqrt 4.9%

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

   2. sqrt-unprod 5.3%

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

   3. sqr-neg 5.3%

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

   4. sqrt-unprod 0.4%

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

   5. *-un-lft-identity 0.4%

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

   6. times-frac 0.4%

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

10. Applied egg-rr 0.4%

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

    1. /-rgt-identity 0.4%

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

    2. associate-*r/ 0.4%

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

    3. rem-square-sqrt 2.6%

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

12. Simplified 2.6%

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

13. Final simplification 2.6%

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

Alternative 7: 27.0% 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 52.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 25.4%

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

5. Final simplification 25.4%

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

Reproduce

herbie shell --seed 2023194 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))