powComplex, imaginary part

Average Error: 33.9 → 18.5

Time: 30.4s

Precision: binary64

Cost: 40396

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{\log x.re \cdot y.re - t_2}\\ t_4 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_2} \cdot t_0\\ \mathbf{if}\;x.re \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - t_2} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-180}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq -9.5 \cdot 10^{-270}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_2} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.re\right) - \frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot 2}{\frac{-1}{y.re}}\right)\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{-219}:\\ \;\;\;\;t_0 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{-189}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;t_3 \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin \left(\log x.re \cdot y.im + t_1\right)\\ \end{array} \]

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 (* y.re (atan2 x.im x.re)))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (exp (- (* (log x.re) y.re) t_2)))
        (t_4
         (*
          (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_2))
          t_0)))
   (if (<= x.re -8.5e-9)
     (*
      (exp (- (* y.re (log (- x.re))) t_2))
      (sin (/ (atan2 x.im x.re) (/ 1.0 y.re))))
     (if (<= x.re -4e-180)
       t_4
       (if (<= x.re -9.5e-270)
         (*
          (exp (- (* (log (* x.re -1.0)) y.re) t_2))
          (sin
           (-
            (* (atan2 x.im x.re) (- y.re))
            (/ (* (atan2 x.im x.re) 2.0) (/ -1.0 y.re)))))
         (if (<= x.re 2.65e-219)
           (* t_0 (exp (- (* y.im (atan2 x.im x.re)))))
           (if (<= x.re 8e-189)
             t_4
             (if (<= x.re 1.3e-162)
               (* t_3 (sin (+ (* (log (- x.im)) y.im) t_1)))
               (* t_3 (sin (+ (* (log x.re) y.im) 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp(((log(x_46_re) * y_46_re) - t_2));
	double t_4 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_2)) * t_0;
	double tmp;
	if (x_46_re <= -8.5e-9) {
		tmp = exp(((y_46_re * log(-x_46_re)) - t_2)) * sin((atan2(x_46_im, x_46_re) / (1.0 / y_46_re)));
	} else if (x_46_re <= -4e-180) {
		tmp = t_4;
	} else if (x_46_re <= -9.5e-270) {
		tmp = exp(((log((x_46_re * -1.0)) * y_46_re) - t_2)) * sin(((atan2(x_46_im, x_46_re) * -y_46_re) - ((atan2(x_46_im, x_46_re) * 2.0) / (-1.0 / y_46_re))));
	} else if (x_46_re <= 2.65e-219) {
		tmp = t_0 * exp(-(y_46_im * atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= 8e-189) {
		tmp = t_4;
	} else if (x_46_re <= 1.3e-162) {
		tmp = t_3 * sin(((log(-x_46_im) * y_46_im) + t_1));
	} else {
		tmp = t_3 * sin(((log(x_46_re) * y_46_im) + t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * sin(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

↓

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = atan2(x_46im, x_46re) * y_46re
    t_2 = atan2(x_46im, x_46re) * y_46im
    t_3 = exp(((log(x_46re) * y_46re) - t_2))
    t_4 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - t_2)) * t_0
    if (x_46re <= (-8.5d-9)) then
        tmp = exp(((y_46re * log(-x_46re)) - t_2)) * sin((atan2(x_46im, x_46re) / (1.0d0 / y_46re)))
    else if (x_46re <= (-4d-180)) then
        tmp = t_4
    else if (x_46re <= (-9.5d-270)) then
        tmp = exp(((log((x_46re * (-1.0d0))) * y_46re) - t_2)) * sin(((atan2(x_46im, x_46re) * -y_46re) - ((atan2(x_46im, x_46re) * 2.0d0) / ((-1.0d0) / y_46re))))
    else if (x_46re <= 2.65d-219) then
        tmp = t_0 * exp(-(y_46im * atan2(x_46im, x_46re)))
    else if (x_46re <= 8d-189) then
        tmp = t_4
    else if (x_46re <= 1.3d-162) then
        tmp = t_3 * sin(((log(-x_46im) * y_46im) + t_1))
    else
        tmp = t_3 * sin(((log(x_46re) * y_46im) + t_1))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = Math.exp(((Math.log(x_46_re) * y_46_re) - t_2));
	double t_4 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_2)) * t_0;
	double tmp;
	if (x_46_re <= -8.5e-9) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_re)) - t_2)) * Math.sin((Math.atan2(x_46_im, x_46_re) / (1.0 / y_46_re)));
	} else if (x_46_re <= -4e-180) {
		tmp = t_4;
	} else if (x_46_re <= -9.5e-270) {
		tmp = Math.exp(((Math.log((x_46_re * -1.0)) * y_46_re) - t_2)) * Math.sin(((Math.atan2(x_46_im, x_46_re) * -y_46_re) - ((Math.atan2(x_46_im, x_46_re) * 2.0) / (-1.0 / y_46_re))));
	} else if (x_46_re <= 2.65e-219) {
		tmp = t_0 * Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= 8e-189) {
		tmp = t_4;
	} else if (x_46_re <= 1.3e-162) {
		tmp = t_3 * Math.sin(((Math.log(-x_46_im) * y_46_im) + t_1));
	} else {
		tmp = t_3 * Math.sin(((Math.log(x_46_re) * y_46_im) + t_1));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_3 = math.exp(((math.log(x_46_re) * y_46_re) - t_2))
	t_4 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_2)) * t_0
	tmp = 0
	if x_46_re <= -8.5e-9:
		tmp = math.exp(((y_46_re * math.log(-x_46_re)) - t_2)) * math.sin((math.atan2(x_46_im, x_46_re) / (1.0 / y_46_re)))
	elif x_46_re <= -4e-180:
		tmp = t_4
	elif x_46_re <= -9.5e-270:
		tmp = math.exp(((math.log((x_46_re * -1.0)) * y_46_re) - t_2)) * math.sin(((math.atan2(x_46_im, x_46_re) * -y_46_re) - ((math.atan2(x_46_im, x_46_re) * 2.0) / (-1.0 / y_46_re))))
	elif x_46_re <= 2.65e-219:
		tmp = t_0 * math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re)))
	elif x_46_re <= 8e-189:
		tmp = t_4
	elif x_46_re <= 1.3e-162:
		tmp = t_3 * math.sin(((math.log(-x_46_im) * y_46_im) + t_1))
	else:
		tmp = t_3 * math.sin(((math.log(x_46_re) * y_46_im) + 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(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = exp(Float64(Float64(log(x_46_re) * y_46_re) - t_2))
	t_4 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_2)) * t_0)
	tmp = 0.0
	if (x_46_re <= -8.5e-9)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - t_2)) * sin(Float64(atan(x_46_im, x_46_re) / Float64(1.0 / y_46_re))));
	elseif (x_46_re <= -4e-180)
		tmp = t_4;
	elseif (x_46_re <= -9.5e-270)
		tmp = Float64(exp(Float64(Float64(log(Float64(x_46_re * -1.0)) * y_46_re) - t_2)) * sin(Float64(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_re)) - Float64(Float64(atan(x_46_im, x_46_re) * 2.0) / Float64(-1.0 / y_46_re)))));
	elseif (x_46_re <= 2.65e-219)
		tmp = Float64(t_0 * exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (x_46_re <= 8e-189)
		tmp = t_4;
	elseif (x_46_re <= 1.3e-162)
		tmp = Float64(t_3 * sin(Float64(Float64(log(Float64(-x_46_im)) * y_46_im) + t_1)));
	else
		tmp = Float64(t_3 * sin(Float64(Float64(log(x_46_re) * y_46_im) + t_1)));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 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)));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	t_3 = exp(((log(x_46_re) * y_46_re) - t_2));
	t_4 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_2)) * t_0;
	tmp = 0.0;
	if (x_46_re <= -8.5e-9)
		tmp = exp(((y_46_re * log(-x_46_re)) - t_2)) * sin((atan2(x_46_im, x_46_re) / (1.0 / y_46_re)));
	elseif (x_46_re <= -4e-180)
		tmp = t_4;
	elseif (x_46_re <= -9.5e-270)
		tmp = exp(((log((x_46_re * -1.0)) * y_46_re) - t_2)) * sin(((atan2(x_46_im, x_46_re) * -y_46_re) - ((atan2(x_46_im, x_46_re) * 2.0) / (-1.0 / y_46_re))));
	elseif (x_46_re <= 2.65e-219)
		tmp = t_0 * exp(-(y_46_im * atan2(x_46_im, x_46_re)));
	elseif (x_46_re <= 8e-189)
		tmp = t_4;
	elseif (x_46_re <= 1.3e-162)
		tmp = t_3 * sin(((log(-x_46_im) * y_46_im) + t_1));
	else
		tmp = t_3 * sin(((log(x_46_re) * y_46_im) + t_1));
	end
	tmp_2 = 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = 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] - t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x$46$re, -8.5e-9], N[(N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -4e-180], t$95$4, If[LessEqual[x$46$re, -9.5e-270], N[(N[Exp[N[(N[(N[Log[N[(x$46$re * -1.0), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$re)), $MachinePrecision] - N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * 2.0), $MachinePrecision] / N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.65e-219], N[(t$95$0 * N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 8e-189], t$95$4, If[LessEqual[x$46$re, 1.3e-162], N[(t$95$3 * N[Sin[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[Sin[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x.re < -8.5e-9

   1. Initial program 40.0

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

   2. Taylor expanded in y.im around 0 26.2

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

   3. Applied egg-rr 26.2

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

   4. Taylor expanded in x.re around -inf 13.1

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

   5. Simplified 13.1

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

      [Start] 13.1	\[ e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]
      rational.json-simplify-39 [=>] 13.1	\[ e^{\log \color{blue}{\left(x.re \cdot -1\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]

   6. Taylor expanded in y.re around 0 13.1

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

   7. Simplified 13.1

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

      [Start] 13.1	\[ e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]
      rational.json-simplify-39 [=>] 13.1	\[ e^{\color{blue}{y.re \cdot \log \left(-1 \cdot x.re\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]
      rational.json-simplify-39 [=>] 13.1	\[ e^{y.re \cdot \log \color{blue}{\left(x.re \cdot -1\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]
      rational.json-simplify-71 [<=] 13.1	\[ e^{y.re \cdot \log \color{blue}{\left(-x.re\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]

   if -8.5e-9 < x.re < -4.0000000000000001e-180 or 2.6500000000000001e-219 < x.re < 8.00000000000000055e-189

   1. Initial program 19.4

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

   2. Taylor expanded in y.im around 0 19.8

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

   3. Taylor expanded in y.re around 0 20.1

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

   if -4.0000000000000001e-180 < x.re < -9.5000000000000006e-270

   1. Initial program 31.9

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

   2. Taylor expanded in y.im around 0 23.8

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

   3. Applied egg-rr 23.8

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

   4. Taylor expanded in x.re around -inf 23.1

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

   5. Simplified 23.1

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

      [Start] 23.1	\[ e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]
      rational.json-simplify-39 [=>] 23.1	\[ e^{\log \color{blue}{\left(x.re \cdot -1\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right) \]

   6. Applied egg-rr 23.2

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

   if -9.5000000000000006e-270 < x.re < 2.6500000000000001e-219

   1. Initial program 31.6

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

   2. Taylor expanded in y.im around 0 24.6

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

   3. Taylor expanded in x.re around 0 39.1

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

   4. Taylor expanded in y.re around 0 29.9

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

   5. Simplified 29.9

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

      [Start] 29.9	\[ e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
      rational.json-simplify-39 [=>] 29.9	\[ \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if 8.00000000000000055e-189 < x.re < 1.3e-162

   1. Initial program 33.1

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

   2. Taylor expanded in x.re around inf 38.4

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

   3. Taylor expanded in x.im around -inf 40.9

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

   4. Simplified 40.9

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

      [Start] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.im + -0.5 \cdot \frac{{x.re}^{2}}{x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-39 [=>] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\color{blue}{x.im \cdot -1} + -0.5 \cdot \frac{{x.re}^{2}}{x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-18 [=>] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(x.im \cdot -1 + \color{blue}{\frac{-0.5}{\frac{x.im}{{x.re}^{2}}}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-1 [=>] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(x.im \cdot -1 + \color{blue}{\frac{{x.re}^{2}}{\frac{x.im}{-0.5}}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Taylor expanded in x.re around 0 40.9

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

   6. Simplified 40.9

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

      [Start] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-39 [=>] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(x.im \cdot -1\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-71 [<=] 40.9	\[ e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.im\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if 1.3e-162 < x.re

   1. Initial program 36.9

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

   2. Taylor expanded in x.re around inf 38.5

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

   3. Taylor expanded in x.re around inf 16.3

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

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-180}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -9.5 \cdot 10^{-270}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.re\right) - \frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot 2}{\frac{-1}{y.re}}\right)\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{-219}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{-189}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.5
Cost	46600

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{y.re \cdot \log \left(-x.re\right) - t_1} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{if}\;x.re \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq -4.7 \cdot 10^{-111}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_1} \cdot \sin \left(\log \left(0.5 \cdot \frac{{x.re}^{2}}{x.im} + x.im\right) \cdot y.im + t_0\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_1} \cdot \sin \left(\log x.re \cdot y.im + t_0\right)\\ \end{array} \]

Alternative 2
Error	18.8
Cost	46536

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{y.re \cdot \log \left(-x.re\right) - t_1} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{if}\;x.re \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_1} \cdot \sin \left(\log x.im \cdot y.im + t_0\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_1} \cdot \sin \left(\log x.re \cdot y.im + t_0\right)\\ \end{array} \]

Alternative 3
Error	18.5
Cost	40344

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{\log x.re \cdot y.re - t_2}\\ t_4 := e^{y.re \cdot \log \left(-x.re\right) - t_2} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ t_5 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_2} \cdot t_0\\ \mathbf{if}\;x.re \leq -7.3 \cdot 10^{-9}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq -1.26 \cdot 10^{-179}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq -4.8 \cdot 10^{-271}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq 2.7 \cdot 10^{-219}:\\ \;\;\;\;t_0 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 9.5 \cdot 10^{-189}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;t_3 \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin \left(\log x.re \cdot y.im + t_1\right)\\ \end{array} \]

Alternative 4
Error	17.3
Cost	39884

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(-x.re\right) - t_0} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{if}\;x.re \leq -9 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -2.3 \cdot 10^{-179}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_0} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_0} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]

Alternative 5
Error	19.0
Cost	33744

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(-x.re\right) - t_0} \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ t_2 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_0} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -8 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -8 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq -5 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 7400000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_0} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 6
Error	21.7
Cost	33296

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq -3.3 \cdot 10^{-256}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_0} \cdot t_1\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{-209}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;x.re \leq 2.7 \cdot 10^{-200}:\\ \;\;\;\;t_1 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;x.re \leq 5.7 \cdot 10^{-49}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(t_2 \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_0} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 7
Error	20.2
Cost	33224

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{\log x.re \cdot y.re - t_0}\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_0} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 1450000000:\\ \;\;\;\;t_1 \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 8
Error	20.1
Cost	33224

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{\log x.re \cdot y.re - t_0}\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 33000000000:\\ \;\;\;\;t_1 \cdot \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 9
Error	20.1
Cost	33224

\[\begin{array}{l} t_0 := \sin \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{1}{y.re}}\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\log x.re \cdot y.re - t_1}\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - t_1} \cdot t_0\\ \mathbf{elif}\;x.re \leq 14600000000:\\ \;\;\;\;t_2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 10
Error	20.2
Cost	33096

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{\log x.re \cdot y.re - t_0}\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(x.re \cdot -1\right) \cdot y.re - t_0} \cdot t_1\\ \mathbf{elif}\;x.re \leq 1450000000:\\ \;\;\;\;t_2 \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 11
Error	28.4
Cost	20104

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;t_0 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	28.4
Cost	20104

\[\begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+59}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	36.5
Cost	19912

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	36.5
Cost	13512

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -9.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	51.0
Cost	6656

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

Reproduce

herbie shell --seed 2023066 
(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)))))