math.log10 on complex, real part

Percentage Accurate: 52.1% → 99.7%

Time: 12.3s

Alternatives: 6

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: 52.1% 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 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing
5. 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 98.6%

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

   4. frac-2neg 98.6%

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

   5. metadata-eval 98.6%

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

   6. neg-log 98.8%

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

   7. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

   1. metadata-eval 98.8%

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

   2. metadata-eval 98.8%

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

   3. neg-log 98.6%

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

   4. frac-2neg 98.6%

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

   5. inv-pow 98.6%

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

   6. metadata-eval 98.6%

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

   7. pow-prod-up 99.8%

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

   8. pow2 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. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\log 10}^{-0.5}\right)}^{2}}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (pow (pow (log 10.0) -0.5) 2.0) (/ 1.0 (log (hypot im re)))))

C

double code(double re, double im) {
	return pow(pow(log(10.0), -0.5), 2.0) / (1.0 / log(hypot(im, re)));
}

Java

public static double code(double re, double im) {
	return Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0) / (1.0 / Math.log(Math.hypot(im, re)));
}

Python

def code(re, im):
	return math.pow(math.pow(math.log(10.0), -0.5), 2.0) / (1.0 / math.log(math.hypot(im, re)))

Julia

function code(re, im)
	return Float64(((log(10.0) ^ -0.5) ^ 2.0) / Float64(1.0 / log(hypot(im, re))))
end

MATLAB

function tmp = code(re, im)
	tmp = ((log(10.0) ^ -0.5) ^ 2.0) / (1.0 / log(hypot(im, re)));
end

Wolfram

code[re_, im_] := N[(N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision] / N[(1.0 / N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing
5. 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 99.1%

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

   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%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.2%

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

   3. associate-/l* 99.6%

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

   4. hypot-undefine 48.3%

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

   5. unpow2 48.3%

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

   6. unpow2 48.3%

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

   7. +-commutative 48.3%

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

   8. unpow2 48.3%

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

   9. unpow2 48.3%

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

   10. hypot-define 99.6%

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

8. Simplified 99.6%

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

   1. associate-*r/ 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. times-frac 99.6%

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

   4. hypot-undefine 48.3%

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

   5. +-commutative 48.3%

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

   6. hypot-undefine 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. add-sqr-sqrt 99.6%

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

   9. clear-num 99.5%

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

   10. associate-*l/ 99.5%

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

   11. *-un-lft-identity 99.5%

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

10. Applied egg-rr 98.9%

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

    1. metadata-eval 98.8%

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

    2. metadata-eval 98.8%

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

    3. neg-log 98.6%

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

    4. frac-2neg 98.6%

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

    5. inv-pow 98.6%

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

    6. metadata-eval 98.6%

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

    7. pow-prod-up 99.8%

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

    8. pow2 99.8%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

    \[\leadsto \frac{{\left({\log 10}^{-0.5}\right)}^{2}}{\frac{1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \]
14. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing
5. 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 99.1%

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

   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%

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

6. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l/ 99.2%

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

   3. associate-/l* 99.6%

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

   4. hypot-undefine 48.3%

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

   5. unpow2 48.3%

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

   6. unpow2 48.3%

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

   7. +-commutative 48.3%

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

   8. unpow2 48.3%

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

   9. unpow2 48.3%

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

   10. hypot-define 99.6%

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

8. Simplified 99.6%

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

   1. pow1/2 99.6%

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

   2. pow-div 98.6%

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

   3. metadata-eval 98.6%

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

   4. inv-pow 98.6%

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

   5. un-div-inv 99.1%

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

   6. hypot-undefine 48.1%

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

   7. +-commutative 48.1%

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

   8. hypot-undefine 99.1%

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

   9. add-sqr-sqrt 99.1%

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

   10. add-sqr-sqrt 99.1%

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

   11. clear-num 99.2%

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

   12. hypot-undefine 48.1%

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

   13. +-commutative 48.1%

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

   14. hypot-undefine 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

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 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing

5. Final simplification 99.1%

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

Alternative 5: 26.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (log(10.0d0) / log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing

5. Taylor expanded in re around 0 24.4%

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

   1. clear-num 24.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   2. inv-pow 24.4%

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

7. Applied egg-rr 24.4%

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

   1. unpow-1 24.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

9. Simplified 24.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

10. Final simplification 24.4%

    \[\leadsto \frac{1}{\frac{\log 10}{\log im}} \]
11. Add Preprocessing

Alternative 6: 26.5% 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 48.1%

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

   1. +-commutative 48.1%

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

   2. +-commutative 48.1%

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

   3. sqr-neg 48.1%

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

   4. sqr-neg 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqr-neg 48.1%

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

   7. hypot-define 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. Add Preprocessing

5. Taylor expanded in re around 0 24.4%

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

6. Final simplification 24.4%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))