powComplex, imaginary part

Average Error: 33.1 → 21.6

Time: 20.3s

Precision: binary64

Cost: 54019

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5.76352593355014 \cdot 10^{-271}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 9.075984131776858 \cdot 10^{-190}:\\ \;\;\;\;0\\ \mathbf{elif}\;x.re \leq 6.5533082775056296 \cdot 10^{-46}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array}\]

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
 (if (<= x.re -5.76352593355014e-271)
   (*
    (sin (* y.re (atan2 x.im x.re)))
    (exp (- (* y.re (log (- x.re))) (* (atan2 x.im x.re) y.im))))
   (if (<= x.re 9.075984131776858e-190)
     0.0
     (if (<= x.re 6.5533082775056296e-46)
       (*
        (exp
         (-
          (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
          (* (atan2 x.im x.re) y.im)))
        (sin
         (+
          (* y.re (atan2 x.im x.re))
          (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))))
       (*
        (exp
         (-
          (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
          (* (atan2 x.im x.re) y.im)))
        (sin (+ (* y.re (atan2 x.im x.re)) (* y.im (log 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 tmp;
	if (x_46_re <= -5.76352593355014e-271) {
		tmp = sin(y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_re * log(-x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im));
	} else if (x_46_re <= 9.075984131776858e-190) {
		tmp = 0.0;
	} else if (x_46_re <= 6.5533082775056296e-46) {
		tmp = exp((y_46_re * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))));
	} else {
		tmp = exp((y_46_re * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(x_46_re)));
	}
	return tmp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Alternatives

Alternative 1
Error	33.7
Cost	176512

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

Alternative 2
Error	33.3
Cost	163392

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\]

Alternative 3
Error	50.6
Cost	144064

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \cdot \sqrt[3]{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right)\]

Alternative 4
Error	50.2
Cost	143936

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \cdot \sqrt[3]{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}\right)\]

Alternative 5
Error	48.3
Cost	117760

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

Alternative 6
Error	33.5
Cost	78848

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

Alternative 7
Error	38.9
Cost	78592

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{{\sin \left(y.im \cdot \log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}\]

Alternative 8
Error	48.9
Cost	78528

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

Alternative 9
Error	44.3
Cost	65536

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

Alternative 10
Error	44.1
Cost	59136

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

Alternative 11
Error	44.4
Cost	59008

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

Alternative 12
Error	48.3
Cost	53120

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\]

Alternative 13
Error	33.1
Cost	53056

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

Alternative 14
Error	36.0
Cost	52992

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

Alternative 15
Error	43.6
Cost	46400

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

Alternative 16
Error	42.8
Cost	46400

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

Alternative 17
Error	42.8
Cost	46336

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

Alternative 18
Error	43.6
Cost	46336

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

Alternative 19
Error	49.9
Cost	46272

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

Alternative 20
Error	50.0
Cost	46272

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

Alternative 21
Error	43.3
Cost	46272

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

Alternative 22
Error	43.7
Cost	46272

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

Alternative 23
Error	26.5
Cost	39616

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

Alternative 24
Error	26.6
Cost	33216

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

Alternative 25
Error	41.8
Cost	32896

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

Alternative 26
Error	43.7
Cost	32896

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

Alternative 27
Error	45.1
Cost	32832

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

Alternative 28
Error	44.0
Cost	32832

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

Alternative 29
Error	61.4
Cost	64

\[1\]

Alternative 30
Error	27.0
Cost	64

\[0\]

Alternative 31
Error	61.5
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if x.re < -5.76352593355014006e-271

   1. Initial program 31.5

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

   2. Taylor expanded around 0 23.4

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

   3. Simplified 23.4

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

   4. Taylor expanded around -inf 17.8

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

   5. Simplified 17.8

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

   6. Simplified 17.8

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

   if -5.76352593355014006e-271 < x.re < 9.0759841317768579e-190

   1. Initial program 24.2

      \[0\]

   if 9.0759841317768579e-190 < x.re < 6.5533082775056296e-46

   1. Initial program 20.6

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

   2. Simplified 20.6

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

   if 6.5533082775056296e-46 < x.re

   1. Initial program 40.5

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

   2. Taylor expanded around inf 26.6

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

   3. Simplified 26.5

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

   4. Simplified 26.5

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.76352593355014 \cdot 10^{-271}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 9.075984131776858 \cdot 10^{-190}:\\ \;\;\;\;0\\ \mathbf{elif}\;x.re \leq 6.5533082775056296 \cdot 10^{-46}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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)))))