math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.4%

Time: 9.4s

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: 51.2% 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 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. 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: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. 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/ 99.2%

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

   3. *-rgt-identity 99.2%

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

   4. hypot-undefine 50.3%

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

   5. unpow2 50.3%

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

   6. unpow2 50.3%

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

   7. +-commutative 50.3%

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

   8. unpow2 50.3%

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

   9. unpow2 50.3%

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

   10. hypot-define 99.2%

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

8. Simplified 99.2%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

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)) * 3.0)) / Float64(-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 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

   1. 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%

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

   2. pow-flip 99.4%

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

   3. pow1/2 99.4%

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

   4. associate-*l/ 99.2%

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

   5. *-un-lft-identity 99.2%

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

   6. times-frac 99.1%

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

   7. *-un-lft-identity 99.1%

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

   8. add-sqr-sqrt 99.1%

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

   9. frac-2neg 99.1%

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

   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 30.4%

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

    2. pow1/3 30.5%

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

    3. log-pow 30.5%

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

    4. pow3 30.5%

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

    5. log-pow 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

Alternative 5: 26.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. +-commutative 50.2%

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

   2. +-commutative 50.2%

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

   3. sqr-neg 50.2%

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

   4. sqr-neg 50.2%

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

   5. hypot-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in re around 0 21.8%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))