powComplex, imaginary part

Average Error: 33.1 → 15.0

Time: 1.6min

Precision: binary64

Cost: 53268

Math

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

↓

\[\begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t_1\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_4 := \log \left(-x.re\right)\\ t_5 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -4.8 \cdot 10^{-60}:\\ \;\;\;\;e^{t_4 \cdot y.re - t_5} \cdot \sin \left(t_4 \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.re \leq -9 \cdot 10^{-210}:\\ \;\;\;\;e^{\frac{y.re \cdot \log t_0}{2} - t_5} \cdot t_1\\ \mathbf{elif}\;x.re \leq -1.45 \cdot 10^{-288}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{-279}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_5} \cdot \sin \left(\log x.im \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.re \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\sqrt{t_0}\right) \cdot y.re - t_5} \cdot \left(y.im \cdot \left(\log x.re \cdot 1\right) + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_5} \cdot \left(y.im \cdot \left(\log x.re \cdot \cos t_1\right) + t_2\right)\\ \end{array} \]

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 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1))
        (t_3 (* (atan2 x.im x.re) y.re))
        (t_4 (log (- x.re)))
        (t_5 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -4.8e-60)
     (* (exp (- (* t_4 y.re) t_5)) (sin (+ (* t_4 y.im) t_3)))
     (if (<= x.re -9e-210)
       (* (exp (- (/ (* y.re (log t_0)) 2.0) t_5)) t_1)
       (if (<= x.re -1.45e-288)
         (* (atan2 x.im x.re) (/ y.re (exp (* y.im (atan2 x.im x.re)))))
         (if (<= x.re 5.8e-279)
           (*
            (exp (- (* (log x.im) y.re) t_5))
            (sin (+ (* (log x.im) y.im) t_3)))
           (if (<= x.re 2.5e-16)
             (*
              (exp (- (* (log (sqrt t_0)) y.re) t_5))
              (+ (* y.im (* (log x.re) 1.0)) t_2))
             (*
              (exp (- (* (log x.re) y.re) t_5))
              (+ (* y.im (* (log x.re) (cos 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 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double t_3 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_4 = log(-x_46_re);
	double t_5 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -4.8e-60) {
		tmp = exp(((t_4 * y_46_re) - t_5)) * sin(((t_4 * y_46_im) + t_3));
	} else if (x_46_re <= -9e-210) {
		tmp = exp((((y_46_re * log(t_0)) / 2.0) - t_5)) * t_1;
	} else if (x_46_re <= -1.45e-288) {
		tmp = atan2(x_46_im, x_46_re) * (y_46_re / exp((y_46_im * atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 5.8e-279) {
		tmp = exp(((log(x_46_im) * y_46_re) - t_5)) * sin(((log(x_46_im) * y_46_im) + t_3));
	} else if (x_46_re <= 2.5e-16) {
		tmp = exp(((log(sqrt(t_0)) * y_46_re) - t_5)) * ((y_46_im * (log(x_46_re) * 1.0)) + t_2);
	} else {
		tmp = exp(((log(x_46_re) * y_46_re) - t_5)) * ((y_46_im * (log(x_46_re) * cos(t_1))) + t_2);
	}
	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) :: t_5
    real(8) :: tmp
    t_0 = (x_46re * x_46re) + (x_46im * x_46im)
    t_1 = y_46re * atan2(x_46im, x_46re)
    t_2 = sin(t_1)
    t_3 = atan2(x_46im, x_46re) * y_46re
    t_4 = log(-x_46re)
    t_5 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-4.8d-60)) then
        tmp = exp(((t_4 * y_46re) - t_5)) * sin(((t_4 * y_46im) + t_3))
    else if (x_46re <= (-9d-210)) then
        tmp = exp((((y_46re * log(t_0)) / 2.0d0) - t_5)) * t_1
    else if (x_46re <= (-1.45d-288)) then
        tmp = atan2(x_46im, x_46re) * (y_46re / exp((y_46im * atan2(x_46im, x_46re))))
    else if (x_46re <= 5.8d-279) then
        tmp = exp(((log(x_46im) * y_46re) - t_5)) * sin(((log(x_46im) * y_46im) + t_3))
    else if (x_46re <= 2.5d-16) then
        tmp = exp(((log(sqrt(t_0)) * y_46re) - t_5)) * ((y_46im * (log(x_46re) * 1.0d0)) + t_2)
    else
        tmp = exp(((log(x_46re) * y_46re) - t_5)) * ((y_46im * (log(x_46re) * cos(t_1))) + t_2)
    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 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double t_3 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_4 = Math.log(-x_46_re);
	double t_5 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -4.8e-60) {
		tmp = Math.exp(((t_4 * y_46_re) - t_5)) * Math.sin(((t_4 * y_46_im) + t_3));
	} else if (x_46_re <= -9e-210) {
		tmp = Math.exp((((y_46_re * Math.log(t_0)) / 2.0) - t_5)) * t_1;
	} else if (x_46_re <= -1.45e-288) {
		tmp = Math.atan2(x_46_im, x_46_re) * (y_46_re / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 5.8e-279) {
		tmp = Math.exp(((Math.log(x_46_im) * y_46_re) - t_5)) * Math.sin(((Math.log(x_46_im) * y_46_im) + t_3));
	} else if (x_46_re <= 2.5e-16) {
		tmp = Math.exp(((Math.log(Math.sqrt(t_0)) * y_46_re) - t_5)) * ((y_46_im * (Math.log(x_46_re) * 1.0)) + t_2);
	} else {
		tmp = Math.exp(((Math.log(x_46_re) * y_46_re) - t_5)) * ((y_46_im * (Math.log(x_46_re) * Math.cos(t_1))) + t_2);
	}
	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 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	t_3 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_4 = math.log(-x_46_re)
	t_5 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -4.8e-60:
		tmp = math.exp(((t_4 * y_46_re) - t_5)) * math.sin(((t_4 * y_46_im) + t_3))
	elif x_46_re <= -9e-210:
		tmp = math.exp((((y_46_re * math.log(t_0)) / 2.0) - t_5)) * t_1
	elif x_46_re <= -1.45e-288:
		tmp = math.atan2(x_46_im, x_46_re) * (y_46_re / math.exp((y_46_im * math.atan2(x_46_im, x_46_re))))
	elif x_46_re <= 5.8e-279:
		tmp = math.exp(((math.log(x_46_im) * y_46_re) - t_5)) * math.sin(((math.log(x_46_im) * y_46_im) + t_3))
	elif x_46_re <= 2.5e-16:
		tmp = math.exp(((math.log(math.sqrt(t_0)) * y_46_re) - t_5)) * ((y_46_im * (math.log(x_46_re) * 1.0)) + t_2)
	else:
		tmp = math.exp(((math.log(x_46_re) * y_46_re) - t_5)) * ((y_46_im * (math.log(x_46_re) * math.cos(t_1))) + 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(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_4 = log(Float64(-x_46_re))
	t_5 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -4.8e-60)
		tmp = Float64(exp(Float64(Float64(t_4 * y_46_re) - t_5)) * sin(Float64(Float64(t_4 * y_46_im) + t_3)));
	elseif (x_46_re <= -9e-210)
		tmp = Float64(exp(Float64(Float64(Float64(y_46_re * log(t_0)) / 2.0) - t_5)) * t_1);
	elseif (x_46_re <= -1.45e-288)
		tmp = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / exp(Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (x_46_re <= 5.8e-279)
		tmp = Float64(exp(Float64(Float64(log(x_46_im) * y_46_re) - t_5)) * sin(Float64(Float64(log(x_46_im) * y_46_im) + t_3)));
	elseif (x_46_re <= 2.5e-16)
		tmp = Float64(exp(Float64(Float64(log(sqrt(t_0)) * y_46_re) - t_5)) * Float64(Float64(y_46_im * Float64(log(x_46_re) * 1.0)) + t_2));
	else
		tmp = Float64(exp(Float64(Float64(log(x_46_re) * y_46_re) - t_5)) * Float64(Float64(y_46_im * Float64(log(x_46_re) * cos(t_1))) + t_2));
	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 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	t_3 = atan2(x_46_im, x_46_re) * y_46_re;
	t_4 = log(-x_46_re);
	t_5 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -4.8e-60)
		tmp = exp(((t_4 * y_46_re) - t_5)) * sin(((t_4 * y_46_im) + t_3));
	elseif (x_46_re <= -9e-210)
		tmp = exp((((y_46_re * log(t_0)) / 2.0) - t_5)) * t_1;
	elseif (x_46_re <= -1.45e-288)
		tmp = atan2(x_46_im, x_46_re) * (y_46_re / exp((y_46_im * atan2(x_46_im, x_46_re))));
	elseif (x_46_re <= 5.8e-279)
		tmp = exp(((log(x_46_im) * y_46_re) - t_5)) * sin(((log(x_46_im) * y_46_im) + t_3));
	elseif (x_46_re <= 2.5e-16)
		tmp = exp(((log(sqrt(t_0)) * y_46_re) - t_5)) * ((y_46_im * (log(x_46_re) * 1.0)) + t_2);
	else
		tmp = exp(((log(x_46_re) * y_46_re) - t_5)) * ((y_46_im * (log(x_46_re) * cos(t_1))) + t_2);
	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[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$4 = N[Log[(-x$46$re)], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -4.8e-60], N[(N[Exp[N[(N[(t$95$4 * y$46$re), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$4 * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -9e-210], N[(N[Exp[N[(N[(N[(y$46$re * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x$46$re, -1.45e-288], N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.8e-279], N[(N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.5e-16], N[(N[Exp[N[(N[(N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x.re < -4.80000000000000019e-60

   1. Initial program 35.4

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

   2. Taylor expanded in x.re around -inf 36.4

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

   3. Simplified 36.4

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

      [Start] 36.4	\[ e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational_best-simplify-1 [=>] 36.4	\[ e^{\log \color{blue}{\left(x.re \cdot -1\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational_best-simplify-10 [=>] 36.4	\[ e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Taylor expanded in x.re around -inf 5.4

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

   5. Simplified 5.4

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

      [Start] 5.4	\[ e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational_best-simplify-1 [=>] 5.4	\[ e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(x.re \cdot -1\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational_best-simplify-10 [=>] 5.4	\[ e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if -4.80000000000000019e-60 < x.re < -9.00000000000000039e-210

   1. Initial program 22.0

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

   2. Taylor expanded in y.im around 0 19.9

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

   3. Taylor expanded in y.re around 0 20.1

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

   4. Applied egg-rr 20.1

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

   if -9.00000000000000039e-210 < x.re < -1.45000000000000007e-288

   1. Initial program 30.4

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

   2. Taylor expanded in y.im around 0 24.3

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

   3. Taylor expanded in x.re around 0 38.5

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

   4. Taylor expanded in y.re around 0 27.4

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

   5. Simplified 27.4

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

      [Start] 27.4	\[ e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
      rational_best-simplify-50 [=>] 27.4	\[ \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)} \]
      rational_best-simplify-1 [<=] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \]
      exponential-simplify-2 [=>] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot \color{blue}{\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\right) \]
      rational_best-simplify-55 [=>] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(1 \cdot \frac{y.re}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)} \]
      rational_best-simplify-1 [=>] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\frac{y.re}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1\right)} \]
      rational_best-simplify-7 [=>] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\frac{y.re}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
      rational_best-simplify-1 [=>] 27.4	\[ \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if -1.45000000000000007e-288 < x.re < 5.8e-279

   1. Initial program 30.8

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

   2. Taylor expanded in x.re around 0 48.5

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

   3. Taylor expanded in x.re around 0 33.5

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

   if 5.8e-279 < x.re < 2.5000000000000002e-16

   1. Initial program 24.6

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

   2. Taylor expanded in y.im around 0 24.9

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

   3. Simplified 24.9

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

      [Start] 24.9	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]
      rational_best-simplify-3 [=>] 24.9	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
      rational_best-simplify-3 [=>] 24.9	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   4. Taylor expanded in x.re around inf 18.8

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

   5. Taylor expanded in y.re around 0 18.8

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

   if 2.5000000000000002e-16 < x.re

   1. Initial program 43.0

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

   2. Taylor expanded in y.im around 0 43.6

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

   3. Simplified 43.6

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

      [Start] 43.6	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]
      rational_best-simplify-3 [=>] 43.6	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
      rational_best-simplify-3 [=>] 43.6	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(\log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   4. Taylor expanded in x.re around inf 28.6

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

   5. Taylor expanded in x.re around inf 14.9

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

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.8 \cdot 10^{-60}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq -9 \cdot 10^{-210}:\\ \;\;\;\;e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -1.45 \cdot 10^{-288}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{-279}:\\ \;\;\;\;e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(\log x.re \cdot 1\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(\log x.re \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.6
Cost	47064

\[\begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \log \left(-x.re\right)\\ t_4 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_5 := e^{\frac{y.re \cdot \log t_0}{2} - t_4} \cdot t_1\\ t_6 := e^{\log x.re \cdot y.re - t_4}\\ \mathbf{if}\;x.re \leq -1.1 \cdot 10^{-61}:\\ \;\;\;\;e^{t_3 \cdot y.re - t_4} \cdot \sin \left(t_3 \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.re \leq -1 \cdot 10^{-209}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq -1.35 \cdot 10^{-288}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{-279}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_4} \cdot \sin \left(\log x.im \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;t_6 \cdot \sin \left(\log \left(\sqrt{t_0}\right) \cdot y.im + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot t_1\\ \end{array} \]

Alternative 2
Error	18.9
Cost	46600

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \log \left(-x.im\right)\\ t_2 := \log \left(-x.re\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_4 := e^{\log x.im \cdot y.re - t_3}\\ \mathbf{if}\;x.im \leq -1.3 \cdot 10^{+32}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.05 \cdot 10^{-228}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_3} \cdot \sin \left(t_1 \cdot y.im + t_0\right)\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{-262}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.im \leq 1.15 \cdot 10^{-242}:\\ \;\;\;\;t_4 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;x.im \leq 3.6 \cdot 10^{-134}:\\ \;\;\;\;e^{t_2 \cdot y.re - t_3} \cdot \sin \left(t_2 \cdot y.im + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \sin \left(\log x.im \cdot y.im + t_0\right)\\ \end{array} \]

Alternative 3
Error	18.5
Cost	40016

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\log x.im \cdot y.re - t_1} \cdot \sin \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_3 := e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - t_1} \cdot t_0\\ \mathbf{if}\;x.im \leq -1:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_1} \cdot \sin t_0\\ \mathbf{elif}\;x.im \leq 1.85 \cdot 10^{-275}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq 1.25 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq 8 \cdot 10^{-107}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	19.0
Cost	40012

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.re\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_3} \cdot \sin t_0\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{-292}:\\ \;\;\;\;e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - t_3} \cdot t_0\\ \mathbf{elif}\;x.im \leq 3.6 \cdot 10^{-134}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot \sin \left(t_1 \cdot y.im + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_3} \cdot \sin \left(\log x.im \cdot y.im + t_2\right)\\ \end{array} \]

Alternative 5
Error	18.9
Cost	40012

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \log \left(-x.re\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -0.00045:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_3} \cdot t_0\\ \mathbf{elif}\;x.im \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_3} \cdot t_0\\ \mathbf{elif}\;x.im \leq 3 \cdot 10^{-134}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot \sin \left(t_1 \cdot y.im + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_3} \cdot \sin \left(\log x.im \cdot y.im + t_2\right)\\ \end{array} \]

Alternative 6
Error	20.1
Cost	27208

\[\begin{array}{l} t_0 := e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -2.65 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	24.0
Cost	26828

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-166}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t_1} \cdot t_0\\ \mathbf{elif}\;x.re \leq -1.15 \cdot 10^{-209}:\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;x.re \leq 7.8 \cdot 10^{-273}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_1} \cdot t_0\\ \end{array} \]

Alternative 8
Error	26.9
Cost	20040

\[\begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+77}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	36.9
Cost	13512

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 640000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	51.2
Cost	6656

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

Reproduce

herbie shell --seed 2023099 
(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)))))