powComplex, imaginary part

Average Error: 33.9 → 19.8

Time: 13.7s

Precision: binary64

Cost: 125059

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -8.216772974650418 \cdot 10^{+102}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -5.119916187162846 \cdot 10^{-88}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\\ \mathbf{elif}\;x.im \leq -3.627285044465458 \cdot 10^{-281}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \left(\sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}}\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.3031182895715489 \cdot 10^{-142}:\\ \;\;\;\;0\\ \mathbf{elif}\;x.im \leq 4168286414130.5063:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \sqrt{\sqrt{{x.re}^{2} + {x.im}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

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.im -8.216772974650418e+102)
   (*
    (sin (* y.re (atan2 x.im x.re)))
    (exp (- (* y.re (log (- x.im))) (* (atan2 x.im x.re) y.im))))
   (if (<= x.im -5.119916187162846e-88)
     (*
      (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 (/ -1.0 x.im))))))
     (if (<= x.im -3.627285044465458e-281)
       (*
        (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
            (*
             (cbrt (sqrt (+ (pow x.re 2.0) (pow x.im 2.0))))
             (*
              (cbrt (sqrt (+ (pow x.re 2.0) (pow x.im 2.0))))
              (cbrt (sqrt (+ (pow x.re 2.0) (pow x.im 2.0)))))))))))
       (if (<= x.im 1.3031182895715489e-142)
         0.0
         (if (<= x.im 4168286414130.5063)
           (*
            (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 (sqrt (+ (pow x.re 2.0) (pow x.im 2.0))))
                 (sqrt (sqrt (+ (pow x.re 2.0) (pow x.im 2.0))))))))))
           (*
            (sin (* y.re (atan2 x.im x.re)))
            (exp (- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))))))))))

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_im <= -8.216772974650418e+102) {
		tmp = sin(y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_re * log(-x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im));
	} else if (x_46_im <= -5.119916187162846e-88) {
		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(-1.0 / x_46_im)));
	} else if (x_46_im <= -3.627285044465458e-281) {
		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(cbrt(sqrt(pow(x_46_re, 2.0) + pow(x_46_im, 2.0))) * (cbrt(sqrt(pow(x_46_re, 2.0) + pow(x_46_im, 2.0))) * cbrt(sqrt(pow(x_46_re, 2.0) + pow(x_46_im, 2.0)))))));
	} else if (x_46_im <= 1.3031182895715489e-142) {
		tmp = 0.0;
	} else if (x_46_im <= 4168286414130.5063) {
		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(sqrt(pow(x_46_re, 2.0) + pow(x_46_im, 2.0))) * sqrt(sqrt(pow(x_46_re, 2.0) + pow(x_46_im, 2.0))))));
	} else {
		tmp = sin(y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im));
	}
	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	34.2
Cost	163392

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

Alternative 2
Error	33.9
Cost	124096

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

Alternative 3
Error	33.9
Cost	98112

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

Alternative 4
Error	44.9
Cost	97984

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

Alternative 5
Error	50.6
Cost	91328

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

Alternative 6
Error	52.4
Cost	85504

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

Alternative 7
Error	50.7
Cost	78720

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

Alternative 8
Error	51.1
Cost	78656

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

Alternative 9
Error	39.4
Cost	78592

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

Alternative 10
Error	44.9
Cost	78528

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

Alternative 11
Error	33.9
Cost	72512

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

Alternative 12
Error	33.9
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 13
Error	36.5
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 14
Error	29.5
Cost	52480

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

Alternative 15
Error	43.9
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	43.5
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	43.9
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 18
Error	43.5
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 19
Error	44.1
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 20
Error	50.3
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 21
Error	50.6
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 22
Error	43.5
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 23
Error	27.4
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	27.5
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	44.1
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 26
Error	42.4
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 27
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 28
Error	45.2
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 29
Error	61.4
Cost	64

\[1\]

Alternative 30
Error	27.5
Cost	64

\[0\]

Alternative 31
Error	61.4
Cost	64

\[-1\]

Derivation

1. Split input into 6 regimes

2. if x.im < -8.2167729746504178e102

   1. Initial program 51.8

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

   2. Taylor expanded around 0 30.4

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

   3. Simplified 30.4

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

   4. Taylor expanded around -inf 13.0

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

   5. Simplified 13.0

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

   6. Simplified 13.0

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

   if -8.2167729746504178e102 < x.im < -5.1199161871628461e-88

   1. Initial program 18.3

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

   2. Taylor expanded around -inf 15.4

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

   3. Simplified 15.4

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

   4. Simplified 15.4

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

   if -5.1199161871628461e-88 < x.im < -3.6272850444654578e-281

   1. Initial program 28.8

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

   3. Applied add-cube-cbrt_binary64 28.8

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

   4. Simplified 28.8

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

   5. Simplified 28.8

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

   6. Simplified 28.8

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

   if -3.6272850444654578e-281 < x.im < 1.30311828957154891e-142

   1. Initial program 29.9

      \[0\]

   if 1.30311828957154891e-142 < x.im < 4168286414130.50635

   1. Initial program 18.2

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

   3. Applied add-sqr-sqrt_binary64 18.2

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

   4. Simplified 18.2

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

   5. Simplified 18.2

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

   6. Simplified 18.2

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

   if 4168286414130.50635 < x.im

   1. Initial program 42.9

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

   2. Taylor expanded around 0 27.6

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

   3. Simplified 27.6

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

   4. Taylor expanded around 0 15.2

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

   5. Simplified 15.2

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 19.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8.216772974650418 \cdot 10^{+102}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -5.119916187162846 \cdot 10^{-88}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\\ \mathbf{elif}\;x.im \leq -3.627285044465458 \cdot 10^{-281}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \left(\sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \sqrt[3]{\sqrt{{x.re}^{2} + {x.im}^{2}}}\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.3031182895715489 \cdot 10^{-142}:\\ \;\;\;\;0\\ \mathbf{elif}\;x.im \leq 4168286414130.5063:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{\sqrt{{x.re}^{2} + {x.im}^{2}}} \cdot \sqrt{\sqrt{{x.re}^{2} + {x.im}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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)))))