powComplex, imaginary part

Average Error: 33.0 → 4.0

Time: 21.7s

Precision: binary64

Cost: 52228

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := t_2 \cdot y.re\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;e^{t_3 - \left|t_0\right|} \cdot t_1\\ \mathbf{elif}\;y.im \leq 10^{-73}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_2, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_3 - t_0} \cdot t_1\\ \end{array} \]

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 (* (atan2 x.im x.re) y.im))
        (t_1 (sin (* y.im (log (hypot x.im x.re)))))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (* t_2 y.re)))
   (if (<= y.im -1.6e-15)
     (* (exp (- t_3 (fabs t_0))) t_1)
     (if (<= y.im 1e-73)
       (/
        (sin (fma t_2 y.im (* y.re (atan2 x.im x.re))))
        (/ 1.0 (pow (hypot x.re x.im) y.re)))
       (* (exp (- t_3 t_0)) t_1)))))

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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = t_2 * y_46_re;
	double tmp;
	if (y_46_im <= -1.6e-15) {
		tmp = exp((t_3 - fabs(t_0))) * t_1;
	} else if (y_46_im <= 1e-73) {
		tmp = sin(fma(t_2, y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))) / (1.0 / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = exp((t_3 - t_0)) * t_1;
	}
	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))) * 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 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = Float64(t_2 * y_46_re)
	tmp = 0.0
	if (y_46_im <= -1.6e-15)
		tmp = Float64(exp(Float64(t_3 - abs(t_0))) * t_1);
	elseif (y_46_im <= 1e-73)
		tmp = Float64(sin(fma(t_2, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) / Float64(1.0 / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(exp(Float64(t_3 - t_0)) * t_1);
	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[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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e-15], N[(N[Exp[N[(t$95$3 - N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 1e-73], N[(N[Sin[N[(t$95$2 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(t$95$3 - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.6e-15

   1. Initial program 34.5

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

   2. Simplified 7.4

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

      [Start] 34.5	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      hypot-def [=>] 34.5	\[ e^{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      fma-def [=>] 34.5	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]
      hypot-def [=>] 7.4	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]
      *-commutative [=>] 7.4	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Applied egg-rr 7.9

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

   4. Simplified 7.9

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

      [Start] 7.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \sqrt{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      unpow2 [=>] 7.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \sqrt{\color{blue}{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      rem-sqrt-square [=>] 7.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{\left|y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      *-commutative [=>] 7.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right|} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   5. Taylor expanded in y.re around 0 34.7

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

   6. Simplified 7.9

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

      [Start] 34.7	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right|} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 34.7	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right|} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 34.7	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right|} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]
      hypot-def [=>] 7.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right|} \cdot \sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   if -1.6e-15 < y.im < 9.99999999999999997e-74

   1. Initial program 31.4

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

   2. Simplified 0.2

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

      [Start] 31.4	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      exp-diff [=>] 31.4	\[ \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      associate-*l/ [=>] 31.4	\[ \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
      /-rgt-identity [<=] 31.4	\[ \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{1}}} \]
      exp-0 [<=] 31.4	\[ \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\color{blue}{e^{0}}}} \]
      *-commutative [=>] 31.4	\[ \frac{\color{blue}{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{0}}} \]
      associate-/l* [=>] 31.4	\[ \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\frac{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{0}}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

   3. Taylor expanded in y.im around 0 0.2

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

   if 9.99999999999999997e-74 < y.im

   1. Initial program 34.9

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

   2. Simplified 6.5

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

      [Start] 34.9	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      hypot-def [=>] 34.9	\[ e^{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      fma-def [=>] 34.9	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]
      hypot-def [=>] 6.5	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]
      *-commutative [=>] 6.5	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Taylor expanded in y.re around 0 35.3

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

   4. Simplified 7.7

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

      [Start] 35.3	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 35.3	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 35.3	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]
      hypot-def [=>] 7.7	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right|} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 10^{-73}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.6
Cost	58688

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

Alternative 2
Error	3.9
Cost	45961

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-15} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-70}\right):\\ \;\;\;\;e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 3
Error	4.2
Cost	45897

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -250 \lor \neg \left(y.im \leq 2500\right):\\ \;\;\;\;e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 4
Error	4.2
Cost	45896

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := t_2 \cdot y.re\\ \mathbf{if}\;y.im \leq -680:\\ \;\;\;\;e^{t_3 - \left|t_0\right|} \cdot \sin t_1\\ \mathbf{elif}\;y.im \leq 390:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_2, y.im, t_1\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{t_3 - t_0} \cdot t_1\\ \end{array} \]

Alternative 5
Error	15.7
Cost	39620

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq 2.4 \cdot 10^{-44}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - t_0} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0} \cdot \sin \left(t_1 + y.im \cdot \log x.im\right)\\ \end{array} \]

Alternative 6
Error	17.8
Cost	39296

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

Alternative 7
Error	18.0
Cost	32896

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

Alternative 8
Error	20.1
Cost	26372

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin t_0}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 9
Error	26.1
Cost	20105

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -6 \lor \neg \left(y.re \leq 1.55 \cdot 10^{+57}\right):\\ \;\;\;\;t_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 10
Error	41.7
Cost	13248

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

Reproduce

herbie shell --seed 2023039 
(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)))))