powComplex, imaginary part

Average Error: 33.8 → 6.7

Time: 33.2s

Precision: binary64

Cost: 90952

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot t_1\\ \mathbf{if}\;y.re \leq -8.194622231283605 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 560:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}\right)}^{3}}\right)}^{y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_1\right)\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* y.re (atan2 x.im x.re)))
        (t_2
         (*
          (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
          t_1)))
   (if (<= y.re -8.194622231283605e-13)
     t_2
     (if (<= y.re 560.0)
       (*
        (/
         (pow (hypot x.re x.im) y.re)
         (pow (cbrt (pow (cbrt (pow (exp (atan2 x.im x.re)) 3.0)) 3.0)) y.im))
        (sin (fma (log (hypot x.re x.im)) y.im t_1)))
       (if (<= y.re 1.7e+62)
         (* t_1 (exp (* (atan2 x.im x.re) (- y.im))))
         (if (<= y.re 1.9e+95)
           (* (exp (- (* y.re (log x.re)) t_0)) (sin t_1))
           t_2))))))

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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * t_1;
	double tmp;
	if (y_46_re <= -8.194622231283605e-13) {
		tmp = t_2;
	} else if (y_46_re <= 560.0) {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / pow(cbrt(pow(cbrt(pow(exp(atan2(x_46_im, x_46_re)), 3.0)), 3.0)), y_46_im)) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_1));
	} else if (y_46_re <= 1.7e+62) {
		tmp = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (y_46_re <= 1.9e+95) {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0)) * sin(t_1);
	} else {
		tmp = t_2;
	}
	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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * t_1)
	tmp = 0.0
	if (y_46_re <= -8.194622231283605e-13)
		tmp = t_2;
	elseif (y_46_re <= 560.0)
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / (cbrt((cbrt((exp(atan(x_46_im, x_46_re)) ^ 3.0)) ^ 3.0)) ^ y_46_im)) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_1)));
	elseif (y_46_re <= 1.7e+62)
		tmp = Float64(t_1 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	elseif (y_46_re <= 1.9e+95)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)) * sin(t_1));
	else
		tmp = t_2;
	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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -8.194622231283605e-13], t$95$2, If[LessEqual[y$46$re, 560.0], N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Power[N[Power[N[Power[N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+62], N[(t$95$1 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+95], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.19462223128360477e-13 or 1.9e95 < y.re

   1. Initial program 33.3

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

   2. Taylor expanded in y.im around 0 3.1

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

   3. Taylor expanded in y.re around 0 3.3

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

   if -8.19462223128360477e-13 < y.re < 560

   1. Initial program 35.1

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

   2. Simplified 7.2

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
      Proof
      (*.f64 (/.f64 (pow.f64 (hypot.f64 x.re x.im) y.re) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (sin.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (sin.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 68 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re))) (pow.f64 (exp.f64 (atan2.f64 x.im x.re)) y.im)) (sin.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (exp.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (atan2.f64 x.im x.re) y.im)))) (sin.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 3 points increase in error, 5 points decrease in error
      (*.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im)))) (sin.f64 (fma.f64 (log.f64 (hypot.f64 x.re x.im)) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 0 points increase in error, 17 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (fma.f64 (log.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))))) y.im (*.f64 y.re (atan2.f64 x.im x.re))))): 79 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (fma.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im (Rewrite<= *-commutative_binary64 (*.f64 (atan2.f64 x.im x.re) y.re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))): 2 points increase in error, 2 points decrease in error

   3. Applied egg-rr 7.2

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

   4. Applied egg-rr 7.2

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

   if 560 < y.re < 1.70000000000000007e62

   1. Initial program 21.2

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

   2. Taylor expanded in y.im around 0 14.1

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

   3. Taylor expanded in x.re around 0 30.4

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

   4. Taylor expanded in y.re around 0 30.4

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

   5. Taylor expanded in y.re around 0 24.8

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

   if 1.70000000000000007e62 < y.re < 1.9e95

   1. Initial program 20.8

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

   2. Taylor expanded in y.im around 0 11.4

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

   3. Taylor expanded in x.re around inf 33.2

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.194622231283605 \cdot 10^{-13}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 560:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}\right)}^{3}}\right)}^{y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	78088

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot t_1\\ \mathbf{if}\;y.re \leq -8.194622231283605 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 560:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_1\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}}\right)}^{y.im}}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	6.6
Cost	65224

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot t_1\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 560:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_1\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t_0} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.3
Cost	46096

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := t_0 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := t_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -4.978731341218567 \cdot 10^{-8}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t_2\\ \mathbf{elif}\;y.im \leq -2.7389000356467625 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.990557984091333 \cdot 10^{-186}:\\ \;\;\;\;\sin t_2 \cdot \frac{t_0}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	14.4
Cost	45832

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2.7389000356467625 \cdot 10^{-225}:\\ \;\;\;\;\frac{t_0}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}} \cdot t_3\\ \mathbf{elif}\;y.im \leq 5.990557984091333 \cdot 10^{-186}:\\ \;\;\;\;t_0 \cdot \sin t_1\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;t_0 \cdot \sin t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	8.0
Cost	45768

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.7
Cost	33040

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := t_0 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := t_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -2.7389000356467625 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.990557984091333 \cdot 10^{-186}:\\ \;\;\;\;t_0 \cdot \sin t_2\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	22.1
Cost	26376

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	26.7
Cost	20104

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.95 \cdot 10^{+34}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	41.8
Cost	13248

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

Reproduce

herbie shell --seed 2022294 
(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)))))