powComplex, imaginary part

Average Accuracy: 31.3% → 61.0%

Time: 23.1s

Precision: binary64

Cost: 71552

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ e^{t_0 \cdot y.re - \sqrt{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]

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) (sqrt (pow (* y.im (atan2 x.im x.re)) 2.0))))
    (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) - sqrt(pow((y_46_im * atan2(x_46_im, x_46_re)), 2.0)))) * 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) - sqrt((Float64(y_46_im * atan(x_46_im, x_46_re)) ^ 2.0)))) * 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[Sqrt[N[Power[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $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 32.4%

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

2. Simplified 68.8%

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

   [Start] 32.4	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

3. Applied egg-rr 69.0%

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

   [Start] 68.8	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   add-sqr-sqrt [=>] 52.8	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{\sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   sqrt-unprod [=>] 69.0	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{\sqrt{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   pow2 [=>] 69.0	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \sqrt{\color{blue}{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}^{2}}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   *-commutative [=>] 69.0	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \sqrt{{\color{blue}{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}^{2}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

4. Final simplification 69.0%

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

Alternatives

Alternative 1
Accuracy	61.1%
Cost	58688

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

Alternative 2
Accuracy	60.4%
Cost	52425

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-61} \lor \neg \left(y.re \leq 8.2 \cdot 10^{-22}\right):\\ \;\;\;\;e^{t_0 \cdot y.re - \sqrt{{t_1}^{2}}} \cdot \sin t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, t_2\right)\right)}{e^{t_1}}\\ \end{array} \]

Alternative 3
Accuracy	60.8%
Cost	45832

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{t_0 \cdot y.re - t_1}\\ \mathbf{if}\;y.re \leq -1.22 \cdot 10^{-13}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, t_2\right)\right)}{e^{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin t_2\\ \end{array} \]

Alternative 4
Accuracy	49.6%
Cost	39560

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-144}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-237}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin t_1\\ \end{array} \]

Alternative 5
Accuracy	49.7%
Cost	39560

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{t_0 \cdot y.re - t_1}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-144}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, t_2\right)\right)}{t_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin t_2\\ \end{array} \]

Alternative 6
Accuracy	49.5%
Cost	33161

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-145} \lor \neg \left(y.re \leq 3.3 \cdot 10^{-239}\right):\\ \;\;\;\;t_1 \cdot e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	43.5%
Cost	33028

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right)\\ \mathbf{if}\;y.im \leq -0.19:\\ \;\;\;\;\sin t_0 \cdot e^{y.re \cdot \log \left(-x.im\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-218}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 8
Accuracy	43.9%
Cost	32976

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right)\\ t_2 := t_0 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;y.im \leq -22500000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.55 \cdot 10^{-216}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	42.7%
Cost	26441

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -12000 \lor \neg \left(y.im \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;t_0 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	38.3%
Cost	20105

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+19} \lor \neg \left(y.re \leq 4.6 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{t_0}{\frac{1}{{x.im}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 11
Accuracy	28.2%
Cost	19912

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ \mathbf{if}\;y.re \leq -42:\\ \;\;\;\;\frac{t_0}{\frac{1}{{x.im}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 12
Accuracy	28.2%
Cost	13641

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -450 \lor \neg \left(y.re \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t_0}{\frac{1}{{x.im}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0\\ \end{array} \]

Alternative 13
Accuracy	13.9%
Cost	13056

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

Alternative 14
Accuracy	13.8%
Cost	6656

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

Reproduce

herbie shell --seed 2023157 
(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)))))