math.log/2 on complex, imaginary part

Percentage Accurate: 49.7% → 99.5%

Time: 9.0s

Alternatives: 3

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: 49.7% 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 54.3%

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

   1. mul0-rgt 99.4%

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

   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. Final simplification 99.5%

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

Alternative 2: 19.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan^{-1}_* \frac{im}{re}\\ \mathbf{if}\;base \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (let* ((t_0 (- (atan2 im re)))) (if (<= base 1.0) t_0 (/ t_0 -1.0))))

C

double code(double re, double im, double base) {
	double t_0 = -atan2(im, re);
	double tmp;
	if (base <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_0 / -1.0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -atan2(im, re)
    if (base <= 1.0d0) then
        tmp = t_0
    else
        tmp = t_0 / (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double t_0 = -Math.atan2(im, re);
	double tmp;
	if (base <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_0 / -1.0;
	}
	return tmp;
}

Python

def code(re, im, base):
	t_0 = -math.atan2(im, re)
	tmp = 0
	if base <= 1.0:
		tmp = t_0
	else:
		tmp = t_0 / -1.0
	return tmp

Julia

function code(re, im, base)
	t_0 = Float64(-atan(im, re))
	tmp = 0.0
	if (base <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(t_0 / -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	t_0 = -atan2(im, re);
	tmp = 0.0;
	if (base <= 1.0)
		tmp = t_0;
	else
		tmp = t_0 / -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := Block[{t$95$0 = (-N[ArcTan[im / re], $MachinePrecision])}, If[LessEqual[base, 1.0], t$95$0, N[(t$95$0 / -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if base < 1

   1. Initial program 58.2%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   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. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

      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. Taylor expanded in base around inf 99.5%

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

      1. inv-pow 99.5%

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

      2. add-cube-cbrt 99.4%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)}}^{-1}\right)} \]

      3. unpow-prod-down 99.4%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1} \cdot {\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

      4. log-prod 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

      5. pow2 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left({\left(\sqrt[3]{base}\right)}^{2}\right)}}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

   6. Applied egg-rr 99.3%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left({\left(\sqrt[3]{base}\right)}^{2}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]
   7. Step-by-step derivation

      1. log-pow 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

      2. log-pow 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \color{blue}{-1 \cdot \log \left(\sqrt[3]{base}\right)}} \]

      3. distribute-lft-out 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(\log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \log \left(\sqrt[3]{base}\right)\right)}} \]

      4. log-pow 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{base}\right)} + \log \left(\sqrt[3]{base}\right)\right)} \]

      5. distribute-lft1-in 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]

      6. metadata-eval 99.3%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{base}\right)\right)} \]

   8. Simplified 99.3%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 99.3%

         \[\leadsto -1 \cdot \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)} \]

      2. add-sqr-sqrt 98.9%

         \[\leadsto -1 \cdot \frac{1 \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)} \cdot \sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}}} \]

      3. times-frac 99.1%

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{1}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}}\right)} \]

   10. Applied egg-rr 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/l* 0.0%

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

       4. rem-square-sqrt 0.0%

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

       5. *-inverses 14.8%

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

       6. associate-*r/ 14.8%

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

       7. /-rgt-identity 14.8%

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

       8. rem-square-sqrt 20.7%

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

   12. Simplified 20.7%

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

   if 1 < base

   1. Initial program 50.1%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   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. Taylor expanded in base around inf 99.6%

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

      1. inv-pow 99.6%

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

      2. add-cube-cbrt 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)}}^{-1}\right)} \]

      3. unpow-prod-down 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1} \cdot {\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

      4. log-prod 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

      5. pow2 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left({\left(\sqrt[3]{base}\right)}^{2}\right)}}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

   6. Applied egg-rr 99.5%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left({\left(\sqrt[3]{base}\right)}^{2}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]
   7. Step-by-step derivation

      1. log-pow 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

      2. log-pow 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \color{blue}{-1 \cdot \log \left(\sqrt[3]{base}\right)}} \]

      3. distribute-lft-out 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(\log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \log \left(\sqrt[3]{base}\right)\right)}} \]

      4. log-pow 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{base}\right)} + \log \left(\sqrt[3]{base}\right)\right)} \]

      5. distribute-lft1-in 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]

      6. metadata-eval 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{base}\right)\right)} \]

   8. Simplified 99.5%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]
   9. Step-by-step derivation

      1. add-log-exp 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \color{blue}{\log \left(e^{3 \cdot \log \left(\sqrt[3]{base}\right)}\right)}} \]

      2. *-commutative 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \left(e^{\color{blue}{\log \left(\sqrt[3]{base}\right) \cdot 3}}\right)} \]

      3. exp-to-pow 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \color{blue}{\left({\left(\sqrt[3]{base}\right)}^{3}\right)}} \]

      4. pow3 99.5%

         \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)}} \]

      5. add-cube-cbrt 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. add-sqr-sqrt 99.1%

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

      8. associate-/l* 99.1%

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

      9. metadata-eval 99.1%

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

      10. /-rgt-identity 99.1%

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

      11. sqrt-div 99.3%

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

      12. clear-num 99.3%

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

      13. add-exp-log 99.0%

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

      14. exp-neg 99.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 21.9%

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

      17. sqr-neg 21.9%

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

      18. sqrt-unprod 21.9%

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

      19. add-sqr-sqrt 21.9%

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

      20. add-exp-log 21.9%

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

   10. Applied egg-rr 21.9%

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

       1. *-inverses 21.9%

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

   12. Simplified 21.9%

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

4. Final simplification 21.3%

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

Alternative 3: 13.9% accurate, 6.0× 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 Float64(-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 54.3%

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

   1. mul0-rgt 99.4%

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

   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. Taylor expanded in base around inf 99.5%

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

   1. inv-pow 99.5%

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

   2. add-cube-cbrt 99.5%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)}}^{-1}\right)} \]

   3. unpow-prod-down 99.5%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1} \cdot {\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

   4. log-prod 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]

   5. pow2 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left({\color{blue}{\left({\left(\sqrt[3]{base}\right)}^{2}\right)}}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left({\left(\sqrt[3]{base}\right)}^{2}\right)}^{-1}\right) + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)}} \]
7. Step-by-step derivation

   1. log-pow 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right)} + \log \left({\left(\sqrt[3]{base}\right)}^{-1}\right)} \]

   2. log-pow 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \color{blue}{-1 \cdot \log \left(\sqrt[3]{base}\right)}} \]

   3. distribute-lft-out 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(\log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \log \left(\sqrt[3]{base}\right)\right)}} \]

   4. log-pow 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{base}\right)} + \log \left(\sqrt[3]{base}\right)\right)} \]

   5. distribute-lft1-in 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]

   6. metadata-eval 99.4%

      \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{base}\right)\right)} \]

8. Simplified 99.4%

   \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 99.4%

      \[\leadsto -1 \cdot \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)} \]

   2. add-sqr-sqrt 51.4%

      \[\leadsto -1 \cdot \frac{1 \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)} \cdot \sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}}} \]

   3. times-frac 51.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{1}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{-1 \cdot \left(3 \cdot \log \left(\sqrt[3]{base}\right)\right)}}\right)} \]

10. Applied egg-rr 5.3%

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

    1. associate-*r/ 5.3%

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

    2. *-commutative 5.3%

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

    3. associate-/l* 5.3%

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

    4. rem-square-sqrt 4.5%

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

    5. *-inverses 12.2%

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

    6. associate-*r/ 12.2%

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

    7. /-rgt-identity 12.2%

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

    8. rem-square-sqrt 16.0%

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

12. Simplified 16.0%

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

13. Final simplification 16.0%

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

Reproduce

herbie shell --seed 2023171 
(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))))