math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.1%

Time: 9.2s

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: 51.9% 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: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64((log(10.0) ^ -0.5) / Float64(sqrt(log(10.0)) / log(hypot(re, im))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. hypot-def 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.0%

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

   2. un-div-inv 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. hypot-def 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.0%

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

   2. un-div-inv 99.1%

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

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
8. Step-by-step derivation

   1. div-inv 99.0%

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

   2. associate-/r* 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-div 98.5%

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

   5. metadata-eval 98.5%

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

   6. inv-pow 98.5%

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

   7. associate-/l/ 98.9%

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

   8. /-rgt-identity 98.9%

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

   9. frac-2neg 98.9%

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

   10. neg-log 99.0%

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

   11. metadata-eval 99.0%

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

   12. metadata-eval 99.0%

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

   13. clear-num 99.0%

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

   14. div-inv 99.0%

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

   15. clear-num 99.1%

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

   16. associate-/l/ 99.0%

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

   17. div-inv 99.0%

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

   18. associate-*l/ 99.1%

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

9. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. metadata-eval 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. 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. frac-2neg 99.0%

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

   2. div-inv 98.5%

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

   3. neg-log 99.0%

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

   4. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. hypot-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 5: 26.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\log 0.1}}{\frac{1}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-1.0d0) / log(0.1d0)) / (1.0d0 / log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(-1.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. hypot-def 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. metadata-eval 99.1%

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

   2. pow-flip 99.1%

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

   3. pow1/2 99.1%

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

   4. times-frac 99.0%

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

   5. add-sqr-sqrt 99.0%

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

   6. associate-/l* 98.9%

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

   7. metadata-eval 98.9%

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

   8. div-inv 98.9%

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

   9. frac-times 98.5%

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

   10. associate-*r/ 98.5%

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

   11. associate-/r/ 98.5%

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

   12. frac-2neg 98.5%

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

   13. metadata-eval 98.5%

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

   14. /-rgt-identity 98.5%

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

   15. neg-log 99.1%

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

   16. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in re around 0 28.1%

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

9. Final simplification 28.1%

   \[\leadsto \frac{\frac{-1}{\log 0.1}}{\frac{1}{\log im}} \]

Alternative 6: 26.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / (log(0.1d0) / log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. hypot-def 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.0%

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

   2. un-div-inv 99.1%

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

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
8. Step-by-step derivation

   1. div-inv 99.0%

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

   2. associate-/r* 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-div 98.5%

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

   5. metadata-eval 98.5%

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

   6. inv-pow 98.5%

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

   7. associate-/l/ 98.9%

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

   8. /-rgt-identity 98.9%

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

   9. frac-2neg 98.9%

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

   10. neg-log 99.0%

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

   11. metadata-eval 99.0%

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

   12. metadata-eval 99.0%

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

   13. clear-num 99.0%

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

   14. div-inv 99.0%

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

   15. clear-num 99.1%

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

   16. associate-/l/ 99.0%

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

   17. div-inv 99.0%

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

   18. associate-*l/ 99.1%

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

9. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. metadata-eval 99.1%

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

11. Simplified 99.1%

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

12. Taylor expanded in re around 0 28.1%

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

13. Final simplification 28.1%

    \[\leadsto \frac{-1}{\frac{\log 0.1}{\log im}} \]

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 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. +-commutative 50.2%

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

   6. +-commutative 50.2%

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

   7. 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 28.1%

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

5. Final simplification 28.1%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))