math.log/2 on complex, imaginary part

Percentage Accurate: 24.8% → 86.6%

Time: 9.6s

Alternatives: 4

Speedup: 2.6×

• Could not converge on a ground truth

  1. re = 1.7533746046869366e+195
  2. im = -1.7030234586554808e+258
  3. base = -1.7593880419529947e+99

Specification

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/
 (-
  (* (atan2 im re) (log base))
  (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
 (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.atan2(im, re) * Math.log(base)) - (Math.log(Math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 24.8% accurate, 1.0× speedup

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/
 (-
  (* (atan2 im re) (log base))
  (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
 (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.atan2(im, re) * Math.log(base)) - (Math.log(Math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\sqrt{base \cdot base}\right)}\\ \mathbf{if}\;base \leq -7.6 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;base \leq -1.56 \cdot 10^{-162}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{0}{re}}{\log \left(\left|base\right|\right)}\\ \mathbf{elif}\;base \leq 1.6 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\\ \end{array} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (let* ((t_0 (/ (atan2 im re) (log (sqrt (* base base))))))
  (if (<= base -7.6e+146)
    t_0
    (if (<= base -1.56e-162)
      (/ (atan2 0.0 re) (log (fabs base)))
      (if (<= base 1.6e-299) t_0 (/ (atan2 im re) (log base)))))))

C

double code(double re, double im, double base) {
	double t_0 = atan2(im, re) / log(sqrt((base * base)));
	double tmp;
	if (base <= -7.6e+146) {
		tmp = t_0;
	} else if (base <= -1.56e-162) {
		tmp = atan2(0.0, re) / log(fabs(base));
	} else if (base <= 1.6e-299) {
		tmp = t_0;
	} else {
		tmp = atan2(im, re) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(im, re) / log(sqrt((base * base)))
    if (base <= (-7.6d+146)) then
        tmp = t_0
    else if (base <= (-1.56d-162)) then
        tmp = atan2(0.0d0, re) / log(abs(base))
    else if (base <= 1.6d-299) then
        tmp = t_0
    else
        tmp = atan2(im, re) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double t_0 = Math.atan2(im, re) / Math.log(Math.sqrt((base * base)));
	double tmp;
	if (base <= -7.6e+146) {
		tmp = t_0;
	} else if (base <= -1.56e-162) {
		tmp = Math.atan2(0.0, re) / Math.log(Math.abs(base));
	} else if (base <= 1.6e-299) {
		tmp = t_0;
	} else {
		tmp = Math.atan2(im, re) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	t_0 = math.atan2(im, re) / math.log(math.sqrt((base * base)))
	tmp = 0
	if base <= -7.6e+146:
		tmp = t_0
	elif base <= -1.56e-162:
		tmp = math.atan2(0.0, re) / math.log(math.fabs(base))
	elif base <= 1.6e-299:
		tmp = t_0
	else:
		tmp = math.atan2(im, re) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	t_0 = Float64(atan(im, re) / log(sqrt(Float64(base * base))))
	tmp = 0.0
	if (base <= -7.6e+146)
		tmp = t_0;
	elseif (base <= -1.56e-162)
		tmp = Float64(atan(0.0, re) / log(abs(base)));
	elseif (base <= 1.6e-299)
		tmp = t_0;
	else
		tmp = Float64(atan(im, re) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	t_0 = atan2(im, re) / log(sqrt((base * base)));
	tmp = 0.0;
	if (base <= -7.6e+146)
		tmp = t_0;
	elseif (base <= -1.56e-162)
		tmp = atan2(0.0, re) / log(abs(base));
	elseif (base <= 1.6e-299)
		tmp = t_0;
	else
		tmp = atan2(im, re) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := Block[{t$95$0 = N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[N[Sqrt[N[(base * base), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[base, -7.6e+146], t$95$0, If[LessEqual[base, -1.56e-162], N[(N[ArcTan[0.0 / re], $MachinePrecision] / N[Log[N[Abs[base], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[base, 1.6e-299], t$95$0, N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if base < -7.5999999999999996e146 or -1.5600000000000001e-162 < base < 1.6e-299

   1. Initial program 24.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      5. mul0-rgt N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      6. --rgt-identity N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      9. lower-/.f64 49.8%

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      13. +-rgt-identity 49.8%

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

   3. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log base \cdot \log base}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \frac{1}{\log base \cdot \log base}\right)} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\frac{1}{\log base \cdot \log base} \cdot \log base\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{\frac{1}{\log base \cdot \log base}} \cdot \log base\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \log base\right) \]

      9. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{{\log base}^{2}}} \cdot \log base\right) \]

      10. pow-flip N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{{\log base}^{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \log base\right) \]

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot {\log base}^{\color{blue}{-1}} \]

      14. inv-pow N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      16. lower-/.f64 49.9%

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log base}} \]

      4. rgt-mult-inverse N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\log base \cdot \frac{1}{\log base}\right)}}{\log base} \]

      5. mult-flip-rev N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{\log base}{\log base}}}{\log base} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\log base}}}{\log base} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      9. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      12. lift-log.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \color{blue}{\log base}} \]

      13. log-pow-rev N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({base}^{\log base}\right)}} \]

      14. log-pow N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log \left(\left|base\right|\right)}} \]

      15. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base}}{\log \left(\left|base\right|\right)}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left|base\right|\right)}} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\left|base\right|\right)}} \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\sqrt{base \cdot base}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\sqrt{base \cdot base}\right)}} \]

      4. lower-*.f64 55.5%

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\sqrt{\color{blue}{base \cdot base}}\right)} \]

   9. Applied rewrites 55.5%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\sqrt{base \cdot base}\right)}} \]

   if -7.5999999999999996e146 < base < -1.5600000000000001e-162

   1. Initial program 24.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      5. mul0-rgt N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      6. --rgt-identity N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      9. lower-/.f64 49.8%

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      13. +-rgt-identity 49.8%

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

   3. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log base \cdot \log base}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \frac{1}{\log base \cdot \log base}\right)} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\frac{1}{\log base \cdot \log base} \cdot \log base\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{\frac{1}{\log base \cdot \log base}} \cdot \log base\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \log base\right) \]

      9. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{{\log base}^{2}}} \cdot \log base\right) \]

      10. pow-flip N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{{\log base}^{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \log base\right) \]

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot {\log base}^{\color{blue}{-1}} \]

      14. inv-pow N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      16. lower-/.f64 49.9%

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log base}} \]

      4. rgt-mult-inverse N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\log base \cdot \frac{1}{\log base}\right)}}{\log base} \]

      5. mult-flip-rev N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{\log base}{\log base}}}{\log base} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\log base}}}{\log base} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      9. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      12. lift-log.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \color{blue}{\log base}} \]

      13. log-pow-rev N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({base}^{\log base}\right)}} \]

      14. log-pow N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log \left(\left|base\right|\right)}} \]

      15. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base}}{\log \left(\left|base\right|\right)}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left|base\right|\right)}} \]

   8. Taylor expanded in undef-var around zero

      \[\leadsto \frac{\tan^{-1}_* \frac{\color{blue}{0}}{re}}{\log \left(\left|base\right|\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.2%

       \[\leadsto \frac{\tan^{-1}_* \frac{\color{blue}{0}}{re}}{\log \left(\left|base\right|\right)} \]

   if 1.6e-299 < base

   1. Initial program 24.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base}} \]

      2. lower-atan2.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{base}} \]

      3. lower-log.f64 50.0%

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;base \leq 10^{-300}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{0}{re}}{\log \left(\left|base\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\\ \end{array} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (if (<= base 1e-300)
  (/ (atan2 0.0 re) (log (fabs base)))
  (/ (atan2 im re) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (base <= 1e-300) {
		tmp = atan2(0.0, re) / log(fabs(base));
	} else {
		tmp = atan2(im, re) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (base <= 1d-300) then
        tmp = atan2(0.0d0, re) / log(abs(base))
    else
        tmp = atan2(im, re) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (base <= 1e-300) {
		tmp = Math.atan2(0.0, re) / Math.log(Math.abs(base));
	} else {
		tmp = Math.atan2(im, re) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if base <= 1e-300:
		tmp = math.atan2(0.0, re) / math.log(math.fabs(base))
	else:
		tmp = math.atan2(im, re) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (base <= 1e-300)
		tmp = Float64(atan(0.0, re) / log(abs(base)));
	else
		tmp = Float64(atan(im, re) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (base <= 1e-300)
		tmp = atan2(0.0, re) / log(abs(base));
	else
		tmp = atan2(im, re) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[base, 1e-300], N[(N[ArcTan[0.0 / re], $MachinePrecision] / N[Log[N[Abs[base], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if base < 1e-300

   1. Initial program 24.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      5. mul0-rgt N/A

         \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      6. --rgt-identity N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      9. lower-/.f64 49.8%

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      13. +-rgt-identity 49.8%

          \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

   3. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log base \cdot \log base}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \frac{1}{\log base \cdot \log base}\right)} \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\frac{1}{\log base \cdot \log base} \cdot \log base\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{\frac{1}{\log base \cdot \log base}} \cdot \log base\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \log base\right) \]

      9. pow2 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{{\log base}^{2}}} \cdot \log base\right) \]

      10. pow-flip N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{{\log base}^{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \log base\right) \]

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot {\log base}^{\color{blue}{-1}} \]

      14. inv-pow N/A

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      16. lower-/.f64 49.9%

          \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log base}} \]

      4. rgt-mult-inverse N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\log base \cdot \frac{1}{\log base}\right)}}{\log base} \]

      5. mult-flip-rev N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{\log base}{\log base}}}{\log base} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\log base}}}{\log base} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

      9. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

      12. lift-log.f64 N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \color{blue}{\log base}} \]

      13. log-pow-rev N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({base}^{\log base}\right)}} \]

      14. log-pow N/A

          \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log \left(\left|base\right|\right)}} \]

      15. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base}}{\log \left(\left|base\right|\right)}} \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left|base\right|\right)}} \]

   8. Taylor expanded in undef-var around zero

      \[\leadsto \frac{\tan^{-1}_* \frac{\color{blue}{0}}{re}}{\log \left(\left|base\right|\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.2%

       \[\leadsto \frac{\tan^{-1}_* \frac{\color{blue}{0}}{re}}{\log \left(\left|base\right|\right)} \]

   if 1e-300 < base

   1. Initial program 24.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base}} \]

      2. lower-atan2.f64 N/A

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{base}} \]

      3. lower-log.f64 50.0%

         \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.7% accurate, 2.6× speedup

Math

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left|base\right|\right)} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/ (atan2 im re) (log (fabs base))))

C

double code(double re, double im, double base) {
	return atan2(im, re) / log(fabs(base));
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = atan2(im, re) / log(abs(base))
end function

Java

public static double code(double re, double im, double base) {
	return Math.atan2(im, re) / Math.log(Math.abs(base));
}

Python

def code(re, im, base):
	return math.atan2(im, re) / math.log(math.fabs(base))

Julia

function code(re, im, base)
	return Float64(atan(im, re) / log(abs(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = atan2(im, re) / log(abs(base));
end

Wolfram

code[re_, im_, base_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[N[Abs[base], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

   3. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   5. mul0-rgt N/A

      \[\leadsto \left(\tan^{-1}_* \frac{im}{re} \cdot \log base - \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   6. --rgt-identity N/A

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

   9. lower-/.f64 49.8%

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base + 0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0 \cdot 0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

   13. +-rgt-identity 49.8%

       \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\log base \cdot \log base} \cdot \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

3. Applied rewrites 49.8%

   \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base} \cdot \left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\log base \cdot \log base}} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\log base \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot \frac{1}{\log base \cdot \log base} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \log base\right)} \cdot \frac{1}{\log base \cdot \log base} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left(\log base \cdot \frac{1}{\log base \cdot \log base}\right)} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\frac{1}{\log base \cdot \log base} \cdot \log base\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{\frac{1}{\log base \cdot \log base}} \cdot \log base\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \log base\right) \]

   9. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\frac{1}{\color{blue}{{\log base}^{2}}} \cdot \log base\right) \]

   10. pow-flip N/A

       \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \left(\color{blue}{{\log base}^{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \log base\right) \]

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot {\log base}^{\color{blue}{-1}} \]

   14. inv-pow N/A

       \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

   16. lower-/.f64 49.9%

       \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

5. Applied rewrites 49.9%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log base}} \]

   2. lift-/.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log base}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log base}} \]

   4. rgt-mult-inverse N/A

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\log base \cdot \frac{1}{\log base}\right)}}{\log base} \]

   5. mult-flip-rev N/A

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{\log base}{\log base}}}{\log base} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base}{\log base}}}{\log base} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{\log base}}{\log base} \]

   9. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log base}} \]

   12. lift-log.f64 N/A

       \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base \cdot \color{blue}{\log base}} \]

   13. log-pow-rev N/A

       \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({base}^{\log base}\right)}} \]

   14. log-pow N/A

       \[\leadsto \frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base \cdot \log \left(\left|base\right|\right)}} \]

   15. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{\log base \cdot \tan^{-1}_* \frac{im}{re}}{\log base}}{\log \left(\left|base\right|\right)}} \]

7. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\left|base\right|\right)}} \]
8. Add Preprocessing

Alternative 4: 50.0% accurate, 2.6× speedup

Math

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]

FPCore

(FPCore (re im base)
  :precision binary64
  (/ (atan2 im re) (log base)))

C

double code(double re, double im, double base) {
	return atan2(im, re) / log(base);
}

Fortran

real(8) function code(re, im, base)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = atan2(im, re) / log(base)
end function

Java

public static double code(double re, double im, double base) {
	return Math.atan2(im, re) / Math.log(base);
}

Python

def code(re, im, base):
	return math.atan2(im, re) / math.log(base)

Julia

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

MATLAB

function tmp = code(re, im, base)
	tmp = atan2(im, re) / log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log base}} \]

   2. lower-atan2.f64 N/A

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{base}} \]

   3. lower-log.f64 50.0%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base} \]

4. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))