math.log10 on complex, real part

Percentage Accurate: 51.2% → 99.4%

Time: 6.3s

Alternatives: 4

Speedup: 0.8×

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\log 10}^{-0.25}\right)}^{4}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (pow (pow (log 10.0) -0.25) 4.0) (pow (log (hypot im re)) -1.0)))

C

double code(double re, double im) {
	return pow(pow(log(10.0), -0.25), 4.0) / pow(log(hypot(im, re)), -1.0);
}

Java

public static double code(double re, double im) {
	return Math.pow(Math.pow(Math.log(10.0), -0.25), 4.0) / Math.pow(Math.log(Math.hypot(im, re)), -1.0);
}

Python

def code(re, im):
	return math.pow(math.pow(math.log(10.0), -0.25), 4.0) / math.pow(math.log(math.hypot(im, re)), -1.0)

Julia

function code(re, im)
	return Float64(((log(10.0) ^ -0.25) ^ 4.0) / (log(hypot(im, re)) ^ -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = ((log(10.0) ^ -0.25) ^ 4.0) / (log(hypot(im, re)) ^ -1.0);
end

Wolfram

code[re_, im_] := N[(N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.25], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 51.1

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-hypot.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. sqr-pow N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{{\left({\log 10}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{\color{blue}{4}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{{\left({\log 10}^{\color{blue}{\frac{-1}{4}}}\right)}^{\left(2 + 2\right)}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]

   13. metadata-eval 99.4

       \[\leadsto \frac{{\left({\log 10}^{-0.25}\right)}^{\color{blue}{4}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{{\left({\log 10}^{-0.25}\right)}^{4}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]
7. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 3: 25.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* -0.5 (fma (/ re im) (/ re im) (* (- (log im)) -2.0))) (log 0.1)))

C

double code(double re, double im) {
	return (-0.5 * fma((re / im), (re / im), (-log(im) * -2.0))) / log(0.1);
}

Julia

function code(re, im)
	return Float64(Float64(-0.5 * fma(Float64(re / im), Float64(re / im), Float64(Float64(-log(im)) * -2.0))) / log(0.1))
end

Wolfram

code[re_, im_] := N[(N[(-0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((-N[Log[im], $MachinePrecision]) * -2.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 98.9

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

4. Applied rewrites 98.9%

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

   1. remove-double-div N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. unpow-1 N/A

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

   5. lift-pow.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow-1 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. lift-log.f64 N/A

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

   6. log-pow N/A

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

   7. inv-pow N/A

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

   8. lift-hypot.f64 N/A

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. log-pow N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \log \left(im \cdot im + re \cdot re\right)}}{\log \frac{1}{10}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \log \left(im \cdot im + re \cdot re\right)}}{\log \frac{1}{10}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2}} \cdot \log \left(im \cdot im + re \cdot re\right)}{\log \frac{1}{10}} \]

   16. lower-log.f64 N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{\log \left(im \cdot im + re \cdot re\right)}}{\log \frac{1}{10}} \]

   17. +-commutative N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}{\log \frac{1}{10}} \]

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 51.2

       \[\leadsto \frac{-0.5 \cdot \log \left(\mathsf{fma}\left(re, re, \color{blue}{im \cdot im}\right)\right)}{\log 0.1} \]

8. Applied rewrites 51.2%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \log \left(\mathsf{fma}\left(re, re, im \cdot im\right)\right)}}{\log 0.1} \]

9. Taylor expanded in im around inf

   \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)}}{\log \frac{1}{10}} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \left(\frac{re \cdot re}{\color{blue}{im \cdot im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log \frac{1}{10}} \]

    4. times-frac N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \left(\color{blue}{\frac{re}{im} \cdot \frac{re}{im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)}{\log \frac{1}{10}} \]

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. log-rec N/A

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

    11. lower-neg.f64 N/A

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

    12. lower-log.f64 27.7

        \[\leadsto \frac{-0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\color{blue}{\log im}\right) \cdot -2\right)}{\log 0.1} \]

11. Applied rewrites 27.7%

    \[\leadsto \frac{-0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)}}{\log 0.1} \]
12. Add Preprocessing

Alternative 4: 27.3% accurate, 1.1× 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. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. lower-log.f64 29.8

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

5. Applied rewrites 29.8%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))