powComplex, real part

Percentage Accurate: 41.2% → 80.0%

Time: 21.3s

Precision: binary64

Cost: 58692

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

↓

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq 8 \cdot 10^{+30}:\\ \;\;\;\;e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - t_0} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp
   (-
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
    (* (atan2 x.im x.re) y.im)))
  (cos
   (+
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im)
    (* (atan2 x.im x.re) y.re)))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= y.re 8e+30)
     (*
      (exp (- (cbrt (pow (* y.re (log (hypot x.re x.im))) 3.0)) t_0))
      (cos (* y.im (log (hypot x.im x.re)))))
     (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= 8e+30) {
		tmp = exp((cbrt(pow((y_46_re * log(hypot(x_46_re, x_46_im))), 3.0)) - t_0)) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= 8e+30) {
		tmp = Math.exp((Math.cbrt(Math.pow((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))), 3.0)) - t_0)) * Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_re <= 8e+30)
		tmp = Float64(exp(Float64(cbrt((Float64(y_46_re * log(hypot(x_46_re, x_46_im))) ^ 3.0)) - t_0)) * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, 8e+30], N[(N[Exp[N[(N[Power[N[Power[N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 8.0000000000000002e30

   1. Initial program 41.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Simplified 81.7%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
      Step-by-step derivation

      [Start] 41.9%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Applied egg-rr 81.7%

      \[\leadsto e^{\color{blue}{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      Step-by-step derivation

      [Start] 81.7%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      add-cbrt-cube [=>] 81.7%	\[ e^{\color{blue}{\sqrt[3]{\left(\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re\right)\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re\right)}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      pow3 [=>] 81.7%	\[ e^{\sqrt[3]{\color{blue}{{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re\right)}^{3}}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      *-commutative [=>] 81.7%	\[ e^{\sqrt[3]{{\color{blue}{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   4. Taylor expanded in y.im around inf 42.4%

      \[\leadsto e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   5. Simplified 83.0%

      \[\leadsto e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 42.4%	\[ e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 42.4%	\[ e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 42.4%	\[ e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]
      hypot-def [=>] 83.0%	\[ e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   if 8.0000000000000002e30 < y.re

   1. Initial program 42.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 75.6%

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

   3. Taylor expanded in y.re around 0 84.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 8 \cdot 10^{+30}:\\ \;\;\;\;e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.0%
Cost	58692

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq 8 \cdot 10^{+30}:\\ \;\;\;\;e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}^{3}} - t_0} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \end{array} \]

Alternative 2
Accuracy	78.6%
Cost	58760

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \mathbf{if}\;y.re \leq -1:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+30}:\\ \;\;\;\;e^{-{\left(\sqrt[3]{t_0}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	78.6%
Cost	46148

\[\begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -0.00165:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	78.8%
Cost	45897

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-6} \lor \neg \left(y.re \leq 1.25 \cdot 10^{+30}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 5
Accuracy	78.8%
Cost	32905

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.22 \lor \neg \left(y.re \leq 3.4 \cdot 10^{+30}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 6
Accuracy	78.8%
Cost	26761

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.29 \lor \neg \left(y.re \leq 1.25 \cdot 10^{+30}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 7
Accuracy	62.1%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;x.im \leq 4.8 \cdot 10^{-304}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 8
Accuracy	35.8%
Cost	19712

\[e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))