math.log10 on complex, real part

Percentage Accurate: 50.8% → 99.4%

Time: 12.3s

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: 50.8% 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: 99.4% 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 49.3%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. 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. add-cube-cbrt 99.0%

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

   3. log-prod 99.0%

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

   4. pow2 99.0%

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

   5. div-inv 98.6%

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

   6. exp-to-pow 98.7%

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

   7. div-inv 98.5%

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

   8. exp-to-pow 98.5%

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

6. Applied egg-rr 98.5%

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

   1. log-pow 98.5%

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

   2. distribute-lft1-in 98.5%

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

   3. metadata-eval 98.5%

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

   4. *-commutative 98.5%

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

8. Simplified 98.5%

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

   1. pow1/3 98.1%

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

   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%

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

10. Applied egg-rr 99.5%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
11. 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%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. 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. add-cube-cbrt 99.0%

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

   3. log-prod 99.0%

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

   4. pow2 99.0%

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

   5. div-inv 98.6%

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

   6. exp-to-pow 98.7%

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

   7. div-inv 98.5%

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

   8. exp-to-pow 98.5%

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

6. Applied egg-rr 98.5%

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

   1. log-pow 98.5%

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

   2. distribute-lft1-in 98.5%

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

   3. metadata-eval 98.5%

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

   4. *-commutative 98.5%

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

8. Simplified 98.5%

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

   1. pow1/3 98.1%

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

   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%

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

10. Applied egg-rr 99.5%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
11. 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%

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

12. Simplified 99.5%

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

    1. expm1-log1p-u 76.0%

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

    2. expm1-udef 76.0%

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

    3. log-pow 76.0%

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

    4. associate-*l* 76.0%

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

14. Applied egg-rr 76.0%

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

    1. expm1-def 76.1%

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

    2. expm1-log1p 99.4%

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

    3. *-commutative 99.4%

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

16. Simplified 99.4%

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

17. Final simplification 99.4%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.3333333333333333d0) / log(0.1d0)) * (3.0d0 * log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(-0.3333333333333333 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] * N[(3.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. 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. add-cube-cbrt 99.0%

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

   3. log-prod 99.0%

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

   4. pow2 99.0%

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

   5. div-inv 98.6%

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

   6. exp-to-pow 98.7%

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

   7. div-inv 98.5%

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

   8. exp-to-pow 98.5%

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

6. Applied egg-rr 98.5%

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

   1. log-pow 98.5%

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

   2. distribute-lft1-in 98.5%

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

   3. metadata-eval 98.5%

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

   4. *-commutative 98.5%

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

8. Simplified 98.5%

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

   1. pow1/3 98.1%

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

   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%

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

10. Applied egg-rr 99.5%

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{-1}{\log 0.1} \cdot 0.3333333333333333\right)}\right)} \cdot 3 \]
11. 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%

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

12. Simplified 99.5%

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

    1. expm1-log1p-u 76.0%

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

    2. expm1-udef 76.0%

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

    3. log-pow 76.0%

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

    4. associate-*l* 76.0%

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

14. Applied egg-rr 76.0%

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

    1. expm1-def 76.1%

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

    2. expm1-log1p 99.4%

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

    3. *-commutative 99.4%

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

16. Simplified 99.4%

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

17. Taylor expanded in re around 0 29.4%

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

    1. *-commutative 29.4%

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

19. Simplified 29.4%

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

20. Final simplification 29.4%

    \[\leadsto \frac{-0.3333333333333333}{\log 0.1} \cdot \left(3 \cdot \log im\right) \]
21. Add Preprocessing

Alternative 5: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

   1. hypot-def 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in re around 0 29.4%

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

6. Final simplification 29.4%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))