math.log10 on complex, real part

Percentage Accurate: 51.0% → 99.5%

Time: 10.6s

Alternatives: 5

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: 51.0% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3 \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (log (pow (hypot re im) (/ -0.3333333333333333 (log 0.1)))) 3.0))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), (-0.3333333333333333 / log(0.1)))) * 3.0;
}

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(Math.hypot(re, im), (-0.3333333333333333 / Math.log(0.1)))) * 3.0;
}

Python

def code(re, im):
	return math.log(math.pow(math.hypot(re, im), (-0.3333333333333333 / math.log(0.1)))) * 3.0

Julia

function code(re, im)
	return Float64(log((hypot(re, im) ^ Float64(-0.3333333333333333 / log(0.1)))) * 3.0)
end

MATLAB

function tmp = code(re, im)
	tmp = log((hypot(re, im) ^ (-0.3333333333333333 / log(0.1)))) * 3.0;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], N[(-0.3333333333333333 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

   1. +-commutative 52.3%

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

   2. +-commutative 52.3%

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

   3. sqr-neg 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqr-neg 52.3%

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

   6. sqr-neg 52.3%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. add-log-exp 99.1%

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

   2. add-cube-cbrt 99.1%

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

   3. log-prod 99.1%

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

   4. pow2 99.1%

      \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   5. div-inv 98.7%

      \[\leadsto \log \left({\left(\sqrt[3]{e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 10}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   6. exp-to-pow 98.8%

      \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   7. div-inv 98.6%

      \[\leadsto \log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 10}}}}\right) \]

   8. exp-to-pow 98.6%

      \[\leadsto \log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]
6. Step-by-step derivation

   1. log-pow 98.6%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right) \]

   2. distribute-lft1-in 98.6%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]

   3. metadata-eval 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

   1. pow1/3 98.2%

      \[\leadsto \log \color{blue}{\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3 \]

   2. pow-pow 98.6%

      \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]

   3. frac-2neg 98.6%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\color{blue}{\frac{-1}{-\log 10}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   4. metadata-eval 98.6%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-1}}{-\log 10} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   5. neg-log 99.5%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   6. metadata-eval 99.5%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log \color{blue}{0.1}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

9. Applied egg-rr 99.5%

   \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
10. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\frac{-1 \cdot 0.3333333333333333}{\log 0.1}\right)}}\right) \cdot 3 \]

    2. metadata-eval 99.5%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-0.3333333333333333}}{\log 0.1}\right)}\right) \cdot 3 \]

11. Simplified 99.5%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right)} \cdot 3 \]

12. Final simplification 99.5%

    \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3 \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 1000} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 3.0 (log (hypot re im))) (log 1000.0)))

C

double code(double re, double im) {
	return (3.0 * log(hypot(re, im))) / log(1000.0);
}

Java

public static double code(double re, double im) {
	return (3.0 * Math.log(Math.hypot(re, im))) / Math.log(1000.0);
}

Python

def code(re, im):
	return (3.0 * math.log(math.hypot(re, im))) / math.log(1000.0)

Julia

function code(re, im)
	return Float64(Float64(3.0 * log(hypot(re, im))) / log(1000.0))
end

MATLAB

function tmp = code(re, im)
	tmp = (3.0 * log(hypot(re, im))) / log(1000.0);
end

Wolfram

code[re_, im_] := N[(N[(3.0 * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Log[1000.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

   1. +-commutative 52.3%

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

   2. +-commutative 52.3%

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

   3. sqr-neg 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqr-neg 52.3%

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

   6. sqr-neg 52.3%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. add-log-exp 99.1%

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

   2. add-cube-cbrt 99.1%

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

   3. log-prod 99.1%

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

   4. pow2 99.1%

      \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   5. div-inv 98.7%

      \[\leadsto \log \left({\left(\sqrt[3]{e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 10}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   6. exp-to-pow 98.8%

      \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}}\right) \]

   7. div-inv 98.6%

      \[\leadsto \log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\log 10}}}}\right) \]

   8. exp-to-pow 98.6%

      \[\leadsto \log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]
6. Step-by-step derivation

   1. log-pow 98.6%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right) \]

   2. distribute-lft1-in 98.6%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}}\right)} \]

   3. metadata-eval 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

   1. pow1/3 98.2%

      \[\leadsto \log \color{blue}{\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3 \]

   2. pow-pow 98.6%

      \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\log 10} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]

   3. frac-2neg 98.6%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\color{blue}{\frac{-1}{-\log 10}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   4. metadata-eval 98.6%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-1}}{-\log 10} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   5. neg-log 99.5%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

   6. metadata-eval 99.5%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log \color{blue}{0.1}} \cdot 0.3333333333333333\right)}\right) \cdot 3 \]

9. Applied egg-rr 99.5%

   \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
10. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left(\frac{-1 \cdot 0.3333333333333333}{\log 0.1}\right)}}\right) \cdot 3 \]

    2. metadata-eval 99.5%

       \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\color{blue}{-0.3333333333333333}}{\log 0.1}\right)}\right) \cdot 3 \]

11. Simplified 99.5%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right)} \cdot 3 \]
12. Step-by-step derivation

    1. expm1-log1p-u 64.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)\right)} \]

    2. expm1-udef 64.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-0.3333333333333333}{\log 0.1}\right)}\right) \cdot 3\right)} - 1} \]

    3. log-pow 64.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \cdot 3\right)} - 1 \]

    4. associate-*l* 64.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)}\right)} - 1 \]

13. Applied egg-rr 64.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 64.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)\right)\right)} \]

    2. expm1-log1p 99.2%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3\right)} \]

    3. *-commutative 99.2%

       \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]

15. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)} \]
16. Step-by-step derivation

    1. expm1-log1p-u 64.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)} \]

    2. expm1-udef 64.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)} - 1} \]

17. Applied egg-rr 14.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}{\log 1000}\right)} - 1} \]
18. Step-by-step derivation

    1. expm1-def 14.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}{\log 1000}\right)\right)} \]

    2. expm1-log1p 33.2%

       \[\leadsto \color{blue}{\frac{\log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}{\log 1000}} \]

    3. log-pow 99.3%

       \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 1000} \]

19. Simplified 99.3%

    \[\leadsto \color{blue}{\frac{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 1000}} \]

20. Final simplification 99.3%

    \[\leadsto \frac{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 1000} \]

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 52.3%

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

   1. +-commutative 52.3%

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

   2. +-commutative 52.3%

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

   3. sqr-neg 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqr-neg 52.3%

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

   6. sqr-neg 52.3%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 26.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{3}{\log 1000} \cdot \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (/ 3.0 (log 1000.0)) (log im)))

C

double code(double re, double im) {
	return (3.0 / log(1000.0)) * log(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (3.0d0 / log(1000.0d0)) * log(im)
end function

Java

public static double code(double re, double im) {
	return (3.0 / Math.log(1000.0)) * Math.log(im);
}

Python

def code(re, im):
	return (3.0 / math.log(1000.0)) * math.log(im)

Julia

function code(re, im)
	return Float64(Float64(3.0 / log(1000.0)) * log(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (3.0 / log(1000.0)) * log(im);
end

Wolfram

code[re_, im_] := N[(N[(3.0 / N[Log[1000.0], $MachinePrecision]), $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

   1. +-commutative 52.3%

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

   2. +-commutative 52.3%

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

   3. sqr-neg 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqr-neg 52.3%

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

   6. sqr-neg 52.3%

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

   7. 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 25.9%

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

   1. clear-num 25.9%

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

   2. inv-pow 25.9%

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

6. Applied egg-rr 25.9%

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

   1. unpow-1 25.9%

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

8. Simplified 25.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
9. Step-by-step derivation

   1. associate-/r/ 25.8%

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

   2. metadata-eval 25.8%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot 3}}{\log 10} \cdot \log im \]

   3. associate-*l/ 25.8%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{\log 10} \cdot 3\right)} \cdot \log im \]

   4. clear-num 25.8%

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{\log 10}{0.3333333333333333}}} \cdot 3\right) \cdot \log im \]

   5. associate-*l/ 25.8%

      \[\leadsto \color{blue}{\frac{1 \cdot 3}{\frac{\log 10}{0.3333333333333333}}} \cdot \log im \]

   6. metadata-eval 25.8%

      \[\leadsto \frac{\color{blue}{3}}{\frac{\log 10}{0.3333333333333333}} \cdot \log im \]

   7. add-log-exp 25.8%

      \[\leadsto \frac{3}{\color{blue}{\log \left(e^{\frac{\log 10}{0.3333333333333333}}\right)}} \cdot \log im \]

   8. div-inv 25.8%

      \[\leadsto \frac{3}{\log \left(e^{\color{blue}{\log 10 \cdot \frac{1}{0.3333333333333333}}}\right)} \cdot \log im \]

   9. metadata-eval 25.8%

      \[\leadsto \frac{3}{\log \left(e^{\log 10 \cdot \color{blue}{3}}\right)} \cdot \log im \]

   10. exp-to-pow 26.1%

       \[\leadsto \frac{3}{\log \color{blue}{\left({10}^{3}\right)}} \cdot \log im \]

   11. metadata-eval 26.1%

       \[\leadsto \frac{3}{\log \color{blue}{1000}} \cdot \log im \]

10. Applied egg-rr 26.1%

    \[\leadsto \color{blue}{\frac{3}{\log 1000} \cdot \log im} \]

11. Final simplification 26.1%

    \[\leadsto \frac{3}{\log 1000} \cdot \log im \]

Alternative 5: 26.6% 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 52.3%

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

   1. +-commutative 52.3%

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

   2. +-commutative 52.3%

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

   3. sqr-neg 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqr-neg 52.3%

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

   6. sqr-neg 52.3%

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

   7. 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 25.9%

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

5. Final simplification 25.9%

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

Reproduce

herbie shell --seed 2023298 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))