math.log10 on complex, real part

Percentage Accurate: 50.5% → 99.4%

Time: 6.1s

Alternatives: 7

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: 50.5% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\log 0.1}^{2} \cdot \frac{-1}{{\log 0.1}^{3}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/
  (* (pow (log 0.1) 2.0) (/ -1.0 (pow (log 0.1) 3.0)))
  (pow (log (hypot im re)) -1.0)))

C

double code(double re, double im) {
	return (pow(log(0.1), 2.0) * (-1.0 / pow(log(0.1), 3.0))) / pow(log(hypot(im, re)), -1.0);
}

Java

public static double code(double re, double im) {
	return (Math.pow(Math.log(0.1), 2.0) * (-1.0 / Math.pow(Math.log(0.1), 3.0))) / Math.pow(Math.log(Math.hypot(im, re)), -1.0);
}

Python

def code(re, im):
	return (math.pow(math.log(0.1), 2.0) * (-1.0 / math.pow(math.log(0.1), 3.0))) / math.pow(math.log(math.hypot(im, re)), -1.0)

Julia

function code(re, im)
	return Float64(Float64((log(0.1) ^ 2.0) * Float64(-1.0 / (log(0.1) ^ 3.0))) / (log(hypot(im, re)) ^ -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = ((log(0.1) ^ 2.0) * (-1.0 / (log(0.1) ^ 3.0))) / (log(hypot(im, re)) ^ -1.0);
end

Wolfram

code[re_, im_] := N[(N[(N[Power[N[Log[0.1], $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / N[Power[N[Log[0.1], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 52.7

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

   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 98.6

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

4. Applied rewrites 98.6%

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{-1}{{\log 0.1}^{3}} \cdot {\log 0.1}^{2}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]

7. Final simplification 99.4%

   \[\leadsto \frac{{\log 0.1}^{2} \cdot \frac{-1}{{\log 0.1}^{3}}}{{\log \left(\mathsf{hypot}\left(im, re\right)\right)}^{-1}} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 52.9

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

   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 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 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 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 4: 25.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * fma(Float64(re / im), Float64(re / im), Float64(log(im) * 2.0))) / log(10.0))
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[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 52.9

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

   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 52.9

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

4. Applied rewrites 52.9%

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

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. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 30.4

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

7. Applied rewrites 30.4%

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

8. Final simplification 30.4%

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

Alternative 5: 25.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(N[Log[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 52.9

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

   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 52.9

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

4. Applied rewrites 52.9%

   \[\leadsto \frac{\color{blue}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \cdot 0.5}}{\log 10} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. metadata-eval N/A

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

   11. lift-log.f64 N/A

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

   12. lower-/.f64 52.8

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

   13. lift-*.f64 N/A

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 52.8

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

6. Applied rewrites 52.8%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 30.3

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

9. Applied rewrites 30.3%

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

10. Final simplification 30.3%

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

Alternative 6: 26.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\log 0.1}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-1.0d0) / (log(0.1d0) / log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 31.4

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

5. Applied rewrites 31.4%

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

7. Applied rewrites 31.4%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log im}}} \]
8. Add Preprocessing

Alternative 7: 26.7% 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 52.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 31.4

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

5. Applied rewrites 31.4%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))