math.log10 on complex, real part

Percentage Accurate: 52.1% → 99.4%

Time: 11.6s

Alternatives: 9

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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.7%

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

   5. unpow-prod-down 99.1%

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

   6. inv-pow 99.1%

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

   7. pow1/2 99.1%

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

   8. pow-flip 99.1%

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

   9. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. associate-/r/ 99.4%

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

8. Simplified 99.4%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[re_, im_] := N[(N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

   3. log-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-/l* 99.0%

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

   2. *-commutative 99.0%

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

8. Applied egg-rr 99.0%

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

10. Applied egg-rr 99.0%

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

    1. distribute-frac-neg 99.0%

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

    2. associate-*r/ 99.4%

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

    3. distribute-rgt-neg-in 99.4%

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

    4. distribute-neg-frac 99.4%

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

    5. metadata-eval 99.4%

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

12. Simplified 99.4%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.7%

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

   5. unpow-prod-down 99.1%

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

   6. inv-pow 99.1%

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

   7. pow1/2 99.1%

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

   8. pow-flip 99.1%

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

   9. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. associate-/r/ 99.4%

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

8. Simplified 99.4%

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

   1. metadata-eval 99.4%

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

   2. sqrt-pow2 99.4%

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

   3. inv-pow 99.4%

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

   4. associate-*l/ 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. times-frac 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. add-sqr-sqrt 99.0%

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

   9. frac-2neg 99.0%

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

   10. neg-log 99.0%

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

   11. metadata-eval 99.0%

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

10. Applied egg-rr 99.0%

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

    1. add-cbrt-cube 35.0%

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

    2. pow1/3 35.1%

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

    3. log-pow 35.1%

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

    4. pow3 35.1%

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

    5. log-pow 99.1%

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

12. Applied egg-rr 99.1%

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

13. Final simplification 99.1%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.7%

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

   5. unpow-prod-down 99.1%

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

   6. inv-pow 99.1%

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

   7. pow1/2 99.1%

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

   8. pow-flip 99.1%

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

   9. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. associate-/r/ 99.4%

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

8. Simplified 99.4%

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

   1. metadata-eval 99.4%

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

   2. sqrt-pow2 99.4%

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

   3. inv-pow 99.4%

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

   4. associate-*l/ 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. times-frac 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. add-sqr-sqrt 99.0%

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

   9. frac-2neg 99.0%

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

   10. neg-log 99.0%

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

   11. metadata-eval 99.0%

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

10. Applied egg-rr 99.0%

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

    1. add-cbrt-cube 35.0%

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

    2. pow1/3 35.1%

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

    3. log-pow 35.1%

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

    4. pow3 35.1%

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

    5. log-pow 99.1%

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

12. Applied egg-rr 99.1%

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

    1. distribute-lft-neg-in 99.1%

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

    2. *-un-lft-identity 99.1%

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

    3. metadata-eval 99.1%

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

    4. associate-*r* 99.1%

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

    5. *-commutative 99.1%

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

    6. associate-*r* 99.2%

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

    7. metadata-eval 99.2%

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

    8. associate-*r* 99.1%

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

    9. metadata-eval 99.1%

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

    10. *-un-lft-identity 99.1%

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

14. Applied egg-rr 99.1%

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

15. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. add-log-exp 99.0%

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

   2. div-inv 98.6%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

8. Applied egg-rr 99.0%

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

Alternative 6: 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 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

Alternative 7: 27.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.7%

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

   5. unpow-prod-down 99.1%

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

   6. inv-pow 99.1%

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

   7. pow1/2 99.1%

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

   8. pow-flip 99.1%

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

   9. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. associate-/r/ 99.4%

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

8. Simplified 99.4%

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

   1. metadata-eval 99.4%

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

   2. sqrt-pow2 99.4%

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

   3. inv-pow 99.4%

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

   4. associate-*l/ 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. times-frac 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. add-sqr-sqrt 99.0%

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

   9. frac-2neg 99.0%

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

   10. neg-log 99.0%

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

   11. metadata-eval 99.0%

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

10. Applied egg-rr 99.0%

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

    1. add-cbrt-cube 35.0%

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

    2. pow1/3 35.1%

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

    3. log-pow 35.1%

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

    4. pow3 35.1%

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

    5. log-pow 99.1%

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

12. Applied egg-rr 99.1%

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

13. Taylor expanded in re around 0 24.6%

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

    1. *-commutative 24.6%

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

15. Simplified 24.6%

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

16. Final simplification 24.6%

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

Alternative 8: 27.9% 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 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 24.6%

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

   1. clear-num 24.6%

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

   2. inv-pow 24.6%

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

7. Applied egg-rr 24.6%

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

   1. unpow-1 24.6%

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

9. Simplified 24.6%

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

Alternative 9: 27.9% 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 51.4%

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

   1. +-commutative 51.4%

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

   2. +-commutative 51.4%

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

   3. sqr-neg 51.4%

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

   4. sqr-neg 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqr-neg 51.4%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 24.6%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))