math.log/2 on complex, imaginary part

Percentage Accurate: 49.6% → 99.5%

Time: 12.5s

Alternatives: 4

Speedup: 3.0×

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: 49.6% 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 45.8%

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

   1. div-sub 45.8%

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

   2. mul0-rgt 99.4%

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

   3. div0 99.4%

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

   4. --rgt-identity 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. times-frac 99.5%

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

   8. *-inverses 99.5%

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

   9. *-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. Final simplification 99.5%

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

Alternative 2: 16.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;base \leq 0.55:\\ \;\;\;\;\left|\tan^{-1}_* \frac{im}{re}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{im}{re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= base 0.55) (fabs (atan2 im re)) (/ 1.0 (/ 1.0 (atan2 im re)))))

C

double code(double re, double im, double base) {
	double tmp;
	if (base <= 0.55) {
		tmp = fabs(atan2(im, re));
	} else {
		tmp = 1.0 / (1.0 / 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 <= 0.55d0) then
        tmp = abs(atan2(im, re))
    else
        tmp = 1.0d0 / (1.0d0 / atan2(im, re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (base <= 0.55) {
		tmp = Math.abs(Math.atan2(im, re));
	} else {
		tmp = 1.0 / (1.0 / Math.atan2(im, re));
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if base <= 0.55:
		tmp = math.fabs(math.atan2(im, re))
	else:
		tmp = 1.0 / (1.0 / math.atan2(im, re))
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (base <= 0.55)
		tmp = abs(atan(im, re));
	else
		tmp = Float64(1.0 / Float64(1.0 / atan(im, re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (base <= 0.55)
		tmp = abs(atan2(im, re));
	else
		tmp = 1.0 / (1.0 / atan2(im, re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[base, 0.55], N[Abs[N[ArcTan[im / re], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if base < 0.55000000000000004

   1. Initial program 41.9%

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

      1. div-sub 41.9%

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

      2. mul0-rgt 99.3%

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

      3. div0 99.3%

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

      4. --rgt-identity 99.3%

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

      5. associate-/l* 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-rgt-identity 99.3%

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

      8. associate-/r* 99.3%

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

      9. *-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 99.4%

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

      2. clear-num 99.3%

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

   6. Applied egg-rr 99.3%

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

   8. Applied egg-rr 3.6%

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

      1. *-commutative 3.6%

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

      2. associate-*l/ 3.6%

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

      3. associate-/l* 3.6%

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

      4. *-inverses 3.6%

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

      5. *-rgt-identity 3.6%

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

   10. Simplified 3.6%

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

       1. remove-double-div 3.6%

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

       2. add-sqr-sqrt 2.6%

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

       3. sqrt-unprod 9.0%

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

       4. pow2 9.0%

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

   12. Applied egg-rr 9.0%

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

       1. unpow2 9.0%

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

       2. rem-sqrt-square 9.3%

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

   14. Simplified 9.3%

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

   if 0.55000000000000004 < base

   1. Initial program 49.4%

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

      1. div-sub 49.4%

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

      2. mul0-rgt 99.5%

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

      3. div0 99.5%

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

      4. --rgt-identity 99.5%

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

      5. associate-/l* 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-rgt-identity 99.3%

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

      8. associate-/r* 99.4%

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

      9. *-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 99.4%

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 20.4%

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

      1. *-commutative 20.4%

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

      2. associate-*l/ 20.4%

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

      3. associate-/l* 20.4%

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

      4. *-inverses 20.4%

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

      5. *-rgt-identity 20.4%

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

   10. Simplified 20.4%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;base \leq 0.55:\\ \;\;\;\;\left|\tan^{-1}_* \frac{im}{re}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\tan^{-1}_* \frac{im}{re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 13.5% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (/ 1.0 (/ 1.0 (atan2 im re))))

C

double code(double re, double im, double base) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / atan2(im, re))
end function

Java

public static double code(double re, double im, double base) {
	return 1.0 / (1.0 / Math.atan2(im, re));
}

Python

def code(re, im, base):
	return 1.0 / (1.0 / math.atan2(im, re))

Julia

function code(re, im, base)
	return Float64(1.0 / Float64(1.0 / atan(im, re)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = 1.0 / (1.0 / atan2(im, re));
end

Wolfram

code[re_, im_, base_] := N[(1.0 / N[(1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.8%

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

   1. div-sub 45.8%

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

   2. mul0-rgt 99.4%

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

   3. div0 99.4%

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

   4. --rgt-identity 99.4%

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

   5. associate-/l* 99.3%

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

   6. metadata-eval 99.3%

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

   7. +-rgt-identity 99.3%

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

   8. associate-/r* 99.4%

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

   9. *-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 99.3%

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

6. Applied egg-rr 99.3%

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

8. Applied egg-rr 12.3%

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

   1. *-commutative 12.3%

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

   2. associate-*l/ 12.3%

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

   3. associate-/l* 12.3%

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

   4. *-inverses 12.3%

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

   5. *-rgt-identity 12.3%

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

10. Simplified 12.3%

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

11. Final simplification 12.3%

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

Alternative 4: 13.5% 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 45.8%

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

   1. div-sub 45.8%

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

   2. mul0-rgt 99.4%

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

   3. div0 99.4%

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

   4. --rgt-identity 99.4%

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

   5. associate-/l* 99.3%

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

   6. metadata-eval 99.3%

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

   7. +-rgt-identity 99.3%

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

   8. associate-/r* 99.4%

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

   9. *-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. Taylor expanded in base around inf 99.4%

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

7. Applied egg-rr 12.3%

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

   1. associate-/r/ 12.3%

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

   2. *-inverses 12.3%

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

   3. *-lft-identity 12.3%

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

9. Simplified 12.3%

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

10. Final simplification 12.3%

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

Reproduce

herbie shell --seed 2024043 
(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))))