powComplex, imaginary part

Percentage Accurate: 41.6% → 80.7%

Time: 32.2s

Precision: binary64

Cost: 71552

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ e^{t_0 \cdot y.re - {\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]

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)))
  (sin
   (+
    (* (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 (log (hypot x.re x.im))))
   (*
    (exp (- (* t_0 y.re) (pow (cbrt (* y.im (atan2 x.im x.re))) 3.0)))
    (sin (fma t_0 y.im (* y.re (atan2 x.im x.re)))))))

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))) * sin(((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 = log(hypot(x_46_re, x_46_im));
	return exp(((t_0 * y_46_re) - pow(cbrt((y_46_im * atan2(x_46_im, x_46_re))), 3.0))) * sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
}

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))) * sin(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 = log(hypot(x_46_re, x_46_im))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - (cbrt(Float64(y_46_im * atan(x_46_im, x_46_re))) ^ 3.0))) * sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))))
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[Sin[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[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[Power[N[Power[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 43.9%

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

2. Simplified 83.7%

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

   [Start] 43.9%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

3. Applied egg-rr 83.7%

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

   [Start] 83.7%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   add-cube-cbrt [=>] 83.7%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   pow3 [=>] 83.7%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}^{3}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   *-commutative [=>] 83.7%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - {\left(\sqrt[3]{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{3}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

4. Final simplification 83.7%

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

Alternatives

Alternative 1
Accuracy	80.7%
Cost	71552

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ e^{t_0 \cdot y.re - {\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]

Alternative 2
Accuracy	80.7%
Cost	58688

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \end{array} \]

Alternative 3
Accuracy	76.8%
Cost	52616

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ \mathbf{if}\;y.re \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t_1\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right) \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(t_1 + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos t_0\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	76.6%
Cost	45896

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right)\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-14}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+30}:\\ \;\;\;\;t_1 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	45769

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+50} \lor \neg \left(y.im \leq 7900000000000\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 6
Accuracy	66.2%
Cost	40144

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t_0\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.im \leq -1.26:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-153}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-167}:\\ \;\;\;\;t_0 \cdot t_2\\ \mathbf{elif}\;y.im \leq 7600000000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	56.4%
Cost	33304

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := t_0 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_3 := t_2 \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+231}:\\ \;\;\;\;\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.im}\right)\\ \mathbf{elif}\;y.im \leq -2.45 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.26:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-165}:\\ \;\;\;\;t_2 \cdot t_0\\ \mathbf{elif}\;y.im \leq 11500000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Accuracy	58.4%
Cost	33028

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -1.7 \cdot 10^{-222}:\\ \;\;\;\;\sin t_0 \cdot e^{y.re \cdot \log \left(-x.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t_0 + y.im \cdot \log x.re\right) \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 9
Accuracy	50.5%
Cost	26436

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;\sin t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t_0 + y.im \cdot \log x.re\right) \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 10
Accuracy	46.0%
Cost	26112

\[\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

Alternative 11
Accuracy	34.6%
Cost	19721

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-10} \lor \neg \left(y.re \leq 3.2 \cdot 10^{-50}\right):\\ \;\;\;\;\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	28.5%
Cost	19657

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-295} \lor \neg \left(y.re \leq 2.05 \cdot 10^{-103}\right):\\ \;\;\;\;\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \end{array} \]

Alternative 13
Accuracy	18.8%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x.re \leq 1.15 \cdot 10^{+239}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 14
Accuracy	12.0%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sin \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(-y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 15
Accuracy	9.6%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;x.re \leq 3.9 \cdot 10^{-239}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 16
Accuracy	4.6%
Cost	12992

\[\sin \left(y.im \cdot \log x.im\right) \]

Reproduce

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