math.log/2 on complex, imaginary part

Percentage Accurate: 50.2% → 99.5%

Time: 12.0s

Alternatives: 3

Speedup: 3.0×

• Could not converge on a ground truth

  1. re = 9.14278784611476e+266
  2. im = 4.823862684952657e+48
  3. base = 2.1977429454928796e+108

Specification

Math

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

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)
    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: 50.2% accurate, 1.0× speedup

Math

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

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)
    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: 99.5% accurate, 3.0× speedup

Math

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

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)
    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 51.6%

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

   1. mul0-rgt 99.4%

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

   2. --rgt-identity 99.4%

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

   3. metadata-eval 99.4%

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

   4. +-rgt-identity 99.4%

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

   5. times-frac 99.5%

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

   6. *-inverses 99.5%

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

   7. *-rgt-identity 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 16.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;base \leq 0.12:\\ \;\;\;\;\left|\tan^{-1}_* \frac{im}{re}\right|\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{im}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= base 0.12) (fabs (atan2 im re)) (atan2 im re)))

C

double code(double re, double im, double base) {
	double tmp;
	if (base <= 0.12) {
		tmp = fabs(atan2(im, re));
	} else {
		tmp = atan2(im, re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (base <= 0.12d0) then
        tmp = abs(atan2(im, re))
    else
        tmp = atan2(im, re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (base <= 0.12) {
		tmp = Math.abs(Math.atan2(im, re));
	} else {
		tmp = Math.atan2(im, re);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if base <= 0.12:
		tmp = math.fabs(math.atan2(im, re))
	else:
		tmp = math.atan2(im, re)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (base <= 0.12)
		tmp = abs(atan(im, re));
	else
		tmp = atan(im, re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (base <= 0.12)
		tmp = abs(atan2(im, re));
	else
		tmp = atan2(im, re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[base, 0.12], N[Abs[N[ArcTan[im / re], $MachinePrecision]], $MachinePrecision], N[ArcTan[im / re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if base < 0.12

   1. Initial program 52.3%

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

      1. mul0-rgt 99.3%

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

      2. --rgt-identity 99.3%

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

      3. associate-/l* 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-rgt-identity 99.2%

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

      6. associate-/r* 99.3%

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

      7. *-inverses 99.3%

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

   3. Simplified 99.3%

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

      1. div-inv 99.5%

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

      2. clear-num 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. clear-num 98.8%

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

      2. associate-/r/ 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. clear-num 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. times-frac 0.0%

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

      7. metadata-eval 0.0%

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

      8. sqrt-div 0.0%

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

      9. add-exp-log 0.0%

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

      10. neg-log 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. sqr-neg 0.0%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. add-exp-log 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-*l/ 0.0%

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

       3. associate-/l* 0.0%

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

       4. *-inverses 10.1%

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

       5. *-rgt-identity 10.1%

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

   12. Simplified 10.1%

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

       1. add-sqr-sqrt 7.4%

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

       2. sqrt-unprod 15.7%

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

       3. pow2 15.7%

          \[\leadsto \sqrt{\color{blue}{{\tan^{-1}_* \frac{im}{re}}^{2}}} \]

   14. Applied egg-rr 15.7%

       \[\leadsto \color{blue}{\sqrt{{\tan^{-1}_* \frac{im}{re}}^{2}}} \]
   15. Step-by-step derivation

       1. unpow2 15.7%

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

       2. rem-sqrt-square 15.6%

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

   16. Simplified 15.6%

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

   if 0.12 < base

   1. Initial program 51.0%

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

      1. mul0-rgt 99.5%

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

      2. --rgt-identity 99.5%

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

      3. associate-/l* 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-rgt-identity 99.3%

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

      6. associate-/r* 99.5%

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

      7. *-inverses 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv 99.6%

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

      2. clear-num 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. clear-num 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. add-sqr-sqrt 98.4%

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

      6. times-frac 98.3%

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

      7. metadata-eval 98.3%

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

      8. sqrt-div 98.4%

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

      9. add-exp-log 98.2%

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

      10. neg-log 98.2%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 21.7%

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

      13. sqr-neg 21.7%

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

      14. sqrt-unprod 21.7%

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

      15. add-sqr-sqrt 21.7%

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

      16. add-exp-log 21.7%

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

   10. Applied egg-rr 21.7%

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

       1. *-commutative 21.7%

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

       2. associate-*l/ 21.7%

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

       3. associate-/l* 21.7%

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

       4. *-inverses 21.7%

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

       5. *-rgt-identity 21.7%

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

   12. Simplified 21.7%

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

Alternative 3: 13.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{im}{re} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im, base)
	return atan(im, re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. mul0-rgt 99.4%

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

   2. --rgt-identity 99.4%

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

   3. associate-/l* 99.3%

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

   4. metadata-eval 99.3%

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

   5. +-rgt-identity 99.3%

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

   6. associate-/r* 99.4%

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

   7. *-inverses 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.5%

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

   2. clear-num 98.9%

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

6. Applied egg-rr 98.9%

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

   1. clear-num 98.9%

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

   2. associate-/r/ 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-*l/ 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. clear-num 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. add-sqr-sqrt 48.0%

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

   6. times-frac 48.0%

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

   7. metadata-eval 48.0%

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

   8. sqrt-div 48.1%

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

   9. add-exp-log 47.9%

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

   10. neg-log 47.9%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 10.6%

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

   13. sqr-neg 10.6%

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

   14. sqrt-unprod 10.6%

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

   15. add-sqr-sqrt 10.6%

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

   16. add-exp-log 10.6%

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

10. Applied egg-rr 10.6%

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

    1. *-commutative 10.6%

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

    2. associate-*l/ 10.6%

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

    3. associate-/l* 10.6%

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

    4. *-inverses 15.7%

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

    5. *-rgt-identity 15.7%

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

12. Simplified 15.7%

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

Reproduce

herbie shell --seed 2024186 
(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))))