math.log10 on complex, real part

Percentage Accurate: 51.0% → 99.0%

Time: 7.5s

Alternatives: 8

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: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{-\log 0.1} \]
6. 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 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.0

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

4. Applied rewrites 99.0%

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

Alternative 3: 25.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{-\log 0.1}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)} \cdot -2} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/
  -1.0
  (* (/ (- (log 0.1)) (fma (/ re im) (/ re im) (* (- -2.0) (log im)))) -2.0)))

C

double code(double re, double im) {
	return -1.0 / ((-log(0.1) / fma((re / im), (re / im), (-(-2.0) * log(im)))) * -2.0);
}

Julia

function code(re, im)
	return Float64(-1.0 / Float64(Float64(Float64(-log(0.1)) / fma(Float64(re / im), Float64(re / im), Float64(Float64(-(-2.0)) * log(im)))) * -2.0))
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[((-N[Log[0.1], $MachinePrecision]) / N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((--2.0) * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. lower-log.f64 N/A

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

   11. metadata-eval 54.4

       \[\leadsto \frac{-1}{\frac{\log \color{blue}{0.1}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \]

   12. lift-sqrt.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-hypot.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. metadata-eval N/A

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

   4. neg-log N/A

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

   5. lift-log.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-hypot.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. log-pow N/A

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

   15. times-frac N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 N/A

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

   21. lower-log.f64 54.4

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

   22. lift-+.f64 N/A

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

   23. lift-*.f64 N/A

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

6. Applied rewrites 54.4%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 25.9

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

9. Applied rewrites 25.9%

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

    1. lift-log.f64 N/A

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

    2. metadata-eval N/A

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

    3. neg-log N/A

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

    4. lift-log.f64 N/A

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

    5. lower-neg.f64 26.0

       \[\leadsto \frac{-1}{-2 \cdot \frac{\color{blue}{-\log 0.1}}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)}} \]

11. Applied rewrites 26.0%

    \[\leadsto \frac{-1}{-2 \cdot \frac{\color{blue}{-\log 0.1}}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)}} \]

12. Final simplification 26.0%

    \[\leadsto \frac{-1}{\frac{-\log 0.1}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)} \cdot -2} \]
13. Add Preprocessing

Alternative 4: 25.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (fma (/ re im) (/ re im) (* (- -2.0) (log im)))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * fma((re / im), (re / im), (-(-2.0) * log(im)))) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * fma(Float64(re / im), Float64(re / im), Float64(Float64(-(-2.0)) * log(im)))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((--2.0) * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. pow-to-exp N/A

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

   5. rem-log-exp N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 54.4

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 54.4

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

4. Applied rewrites 54.4%

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

5. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 25.9

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\color{blue}{\log im}\right) \cdot -2\right) \cdot 0.5}{\log 10} \]

7. Applied rewrites 25.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)} \cdot 0.5}{\log 10} \]

8. Final simplification 25.9%

   \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \]
9. Add Preprocessing

Alternative 5: 27.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{-\log 0.1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 27.6

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

5. Applied rewrites 27.6%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. metadata-eval N/A

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

   6. lift-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 27.7

      \[\leadsto \frac{\color{blue}{-\log im}}{\log 0.1} \]

7. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\frac{-\log im}{\log 0.1}} \]

8. Final simplification 27.7%

   \[\leadsto \frac{\log im}{-\log 0.1} \]
9. Add Preprocessing

Alternative 6: 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 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 27.6

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

5. Applied rewrites 27.6%

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

Alternative 7: 3.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 10}{\frac{re}{im} \cdot \frac{re}{im}} \cdot -2} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ -1.0 (* (/ (log 10.0) (* (/ re im) (/ re im))) -2.0)))

C

double code(double re, double im) {
	return -1.0 / ((log(10.0) / ((re / im) * (re / im))) * -2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / ((log(10.0d0) / ((re / im) * (re / im))) * (-2.0d0))
end function

Java

public static double code(double re, double im) {
	return -1.0 / ((Math.log(10.0) / ((re / im) * (re / im))) * -2.0);
}

Python

def code(re, im):
	return -1.0 / ((math.log(10.0) / ((re / im) * (re / im))) * -2.0)

Julia

function code(re, im)
	return Float64(-1.0 / Float64(Float64(log(10.0) / Float64(Float64(re / im) * Float64(re / im))) * -2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0 / ((log(10.0) / ((re / im) * (re / im))) * -2.0);
end

Wolfram

code[re_, im_] := N[(-1.0 / N[(N[(N[Log[10.0], $MachinePrecision] / N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. lower-log.f64 N/A

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

   11. metadata-eval 54.4

       \[\leadsto \frac{-1}{\frac{\log \color{blue}{0.1}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \]

   12. lift-sqrt.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-hypot.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. metadata-eval N/A

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

   4. neg-log N/A

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

   5. lift-log.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-hypot.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. log-pow N/A

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

   15. times-frac N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 N/A

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

   21. lower-log.f64 54.4

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

   22. lift-+.f64 N/A

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

   23. lift-*.f64 N/A

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

6. Applied rewrites 54.4%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 25.9

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

9. Applied rewrites 25.9%

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

10. Taylor expanded in re around inf

    \[\leadsto \frac{-1}{-2 \cdot \frac{\log 10}{\frac{{re}^{2}}{\color{blue}{{im}^{2}}}}} \]
11. Step-by-step derivation

12. Applied rewrites 3.4%

    \[\leadsto \frac{-1}{-2 \cdot \frac{\log 10}{\frac{re}{im} \cdot \color{blue}{\frac{re}{im}}}} \]

13. Final simplification 3.4%

    \[\leadsto \frac{-1}{\frac{\log 10}{\frac{re}{im} \cdot \frac{re}{im}} \cdot -2} \]
14. Add Preprocessing

Alternative 8: 2.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((re * re) * (-0.5d0)) / ((log(0.1d0) * im) * im)
end function

Java

public static double code(double re, double im) {
	return ((re * re) * -0.5) / ((Math.log(0.1) * im) * im);
}

Python

def code(re, im):
	return ((re * re) * -0.5) / ((math.log(0.1) * im) * im)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = ((re * re) * -0.5) / ((log(0.1) * im) * im);
end

Wolfram

code[re_, im_] := N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] / N[(N[(N[Log[0.1], $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-neg.f64 N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. metadata-eval N/A

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

   8. remove-double-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   11. lower-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-neg.f64 N/A

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

   15. remove-double-neg N/A

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

   16. lower-/.f64 99.1

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

   17. lift-hypot.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. +-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. lower-hypot.f64 99.1

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

9. Applied rewrites 26.0%

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

10. Taylor expanded in re around inf

    \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log \frac{1}{10}}} \]
11. Step-by-step derivation

12. Applied rewrites 2.9%

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

13. Final simplification 2.9%

    \[\leadsto \frac{\left(re \cdot re\right) \cdot -0.5}{\left(\log 0.1 \cdot im\right) \cdot im} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))