math.log10 on complex, real part

Percentage Accurate: 51.4% → 99.0%

Time: 9.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return log(hypot(re, im)) / (0.0 - log(0.1));
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

   1. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right) \]

   3. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{1}{10}\right)\right)\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{10}\right)\right)\right)\right) \]

   5. metadata-eval 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{0 - \log 0.1} \]
8. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

Alternative 3: 26.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{re \cdot \frac{re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/
  (log
   (+
    im
    (* re (* re (+ (/ (* -0.125 (/ (* re (/ re im)) im)) im) (/ 0.5 im))))))
  (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / (0.0 - log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im + (re * (re * ((((-0.125d0) * ((re * (re / im)) / im)) / im) + (0.5d0 / im)))))) / (0.0d0 - log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return Math.log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(log(Float64(im + Float64(re * Float64(re * Float64(Float64(Float64(-0.125 * Float64(Float64(re * Float64(re / im)) / im)) / im) + Float64(0.5 / im)))))) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[Log[N[(im + N[(re * N[(re * N[(N[(N[(-0.125 * N[(N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] + N[(0.5 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right), \mathsf{log.f64}\left(10\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(re \cdot \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 23.8%

   \[\leadsto \frac{\log \color{blue}{\left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\frac{re \cdot re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}}{\log 10} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(re \cdot \frac{re}{im}\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{re}{im} \cdot re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{re}{im}\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. /-lowering-/.f64 24.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

9. Applied egg-rr 24.0%

   \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\color{blue}{\frac{re}{im} \cdot re}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]
10. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \log \left(\frac{1}{\frac{1}{10}}\right)\right) \]

    2. neg-log N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\log \frac{1}{10}\right)\right)\right) \]

    3. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{neg.f64}\left(\log \frac{1}{10}\right)\right) \]

    4. log-lowering-log.f64 24.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

11. Applied egg-rr 24.1%

    \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\frac{re}{im} \cdot re}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\color{blue}{-\log 0.1}} \]

12. Final simplification 24.1%

    \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{re \cdot \frac{re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{0 - \log 0.1} \]
13. Add Preprocessing

Alternative 4: 26.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{re \cdot \frac{re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/
  (log
   (+
    im
    (* re (* re (+ (/ (* -0.125 (/ (* re (/ re im)) im)) im) (/ 0.5 im))))))
  (log 10.0)))

C

double code(double re, double im) {
	return log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im + (re * (re * ((((-0.125d0) * ((re * (re / im)) / im)) / im) + (0.5d0 / im)))))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(Float64(im + Float64(re * Float64(re * Float64(Float64(Float64(-0.125 * Float64(Float64(re * Float64(re / im)) / im)) / im) + Float64(0.5 / im)))))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im + (re * (re * (((-0.125 * ((re * (re / im)) / im)) / im) + (0.5 / im)))))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[(im + N[(re * N[(re * N[(N[(N[(-0.125 * N[(N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] + N[(0.5 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right), \mathsf{log.f64}\left(10\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(re \cdot \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 23.8%

   \[\leadsto \frac{\log \color{blue}{\left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\frac{re \cdot re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}}{\log 10} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(re \cdot \frac{re}{im}\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{re}{im} \cdot re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{re}{im}\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. /-lowering-/.f64 24.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

9. Applied egg-rr 24.0%

   \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\color{blue}{\frac{re}{im} \cdot re}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]

10. Final simplification 24.0%

    \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{re \cdot \frac{re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]
11. Add Preprocessing

Alternative 5: 27.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im + (re * (re * (0.5d0 / im))))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log((im + (re * (re * (0.5 / im))))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log((im + (re * (re * (0.5 / im))))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(Float64(im + Float64(re * Float64(re * Float64(0.5 / im))))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im + (re * (re * (0.5 / im))))) / log(10.0);
end

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right), \mathsf{log.f64}\left(10\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(re \cdot \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 23.8%

   \[\leadsto \frac{\log \color{blue}{\left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\frac{re \cdot re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}}{\log 10} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(re \cdot \frac{re}{im}\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{re}{im} \cdot re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{re}{im}\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. /-lowering-/.f64 24.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

9. Applied egg-rr 24.0%

   \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\color{blue}{\frac{re}{im} \cdot re}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]

10. Taylor expanded in re around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot \frac{re}{im}\right)}\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(\frac{\frac{1}{2} \cdot re}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(\frac{re \cdot \frac{1}{2}}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \frac{\frac{1}{2}}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \frac{\frac{1}{2} \cdot 1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    7. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{\frac{1}{2} \cdot 1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{\frac{1}{2}}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

    9. /-lowering-/.f64 25.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

12. Simplified 25.0%

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

Alternative 6: 27.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 24.9%

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

9. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\frac{0 - \log im}{{\log 0.1}^{3}} \cdot {\log 0.1}^{2}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\left(0 - \log im\right) \cdot {\log \frac{1}{10}}^{2}}{\color{blue}{{\log \frac{1}{10}}^{3}}} \]

    2. associate-/l* N/A

       \[\leadsto \left(0 - \log im\right) \cdot \color{blue}{\frac{{\log \frac{1}{10}}^{2}}{{\log \frac{1}{10}}^{3}}} \]

    3. pow-div N/A

       \[\leadsto \left(0 - \log im\right) \cdot {\log \frac{1}{10}}^{\color{blue}{\left(2 - 3\right)}} \]

    4. metadata-eval N/A

       \[\leadsto \left(0 - \log im\right) \cdot {\log \frac{1}{10}}^{-1} \]

    5. inv-pow N/A

       \[\leadsto \left(0 - \log im\right) \cdot \frac{1}{\color{blue}{\log \frac{1}{10}}} \]

    6. div-inv N/A

       \[\leadsto \frac{0 - \log im}{\color{blue}{\log \frac{1}{10}}} \]

    7. sub0-neg N/A

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

    8. distribute-frac-neg N/A

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

    9. distribute-frac-neg2 N/A

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

    10. clear-num N/A

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

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\mathsf{neg}\left(\log \frac{1}{10}\right)}{\log im}\right)}\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\log \frac{1}{10}\right)\right), \color{blue}{\log im}\right)\right) \]

    13. neg-log N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\log \left(\frac{1}{\frac{1}{10}}\right), \log \color{blue}{im}\right)\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\log 10, \log im\right)\right) \]

    15. log-lowering-log.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(10\right), \log \color{blue}{im}\right)\right) \]

    16. log-lowering-log.f64 25.0%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(10\right), \mathsf{log.f64}\left(im\right)\right)\right) \]

11. Applied egg-rr 25.0%

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

Alternative 7: 27.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return log(im) / (0.0 - log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / (0.0d0 - log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.log(im) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(log(im) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[(0.0 - N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 24.9%

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \left(\frac{1}{\frac{1}{10}}\right)\right) \]

   2. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \left(\mathsf{neg}\left(\log \frac{1}{10}\right)\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\log \frac{1}{10}\right)\right) \]

   4. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

9. Applied egg-rr 25.0%

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

10. Final simplification 25.0%

    \[\leadsto \frac{\log im}{0 - \log 0.1} \]
11. Add Preprocessing

Alternative 8: 27.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 24.9%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))