powComplex, imaginary part

Average Error: 33.5 → 17.6

Time: 45.4s

Precision: binary64

Cost: 46800

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-x.im\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \sin \left(\log x.im \cdot y.im + t_2\right)\\ t_4 := e^{\log x.im \cdot y.re - t_1}\\ \mathbf{if}\;x.im \leq -2.25 \cdot 10^{-299}:\\ \;\;\;\;e^{t_0 \cdot y.re - t_1} \cdot \sin \left(t_0 \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.im \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_1} \cdot \sin \left(\log x.re \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.im \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;t_4 \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.im \leq 1.16 \cdot 10^{+14}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t_1} \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot t_3\\ \end{array} \]

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 (log (- x.im)))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* (atan2 x.im x.re) y.re))
        (t_3 (sin (+ (* (log x.im) y.im) t_2)))
        (t_4 (exp (- (* (log x.im) y.re) t_1))))
   (if (<= x.im -2.25e-299)
     (* (exp (- (* t_0 y.re) t_1)) (sin (+ (* t_0 y.im) t_2)))
     (if (<= x.im 3.6e-166)
       (* (exp (- (* (log x.re) y.re) t_1)) (sin (+ (* (log x.re) y.im) t_2)))
       (if (<= x.im 1.4e-124)
         (* t_4 (sin (+ (* (log (- x.re)) y.im) t_2)))
         (if (<= x.im 1.16e+14)
           (*
            (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_1))
            t_3)
           (* t_4 t_3)))))))

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 = log(-x_46_im);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_3 = sin(((log(x_46_im) * y_46_im) + t_2));
	double t_4 = exp(((log(x_46_im) * y_46_re) - t_1));
	double tmp;
	if (x_46_im <= -2.25e-299) {
		tmp = exp(((t_0 * y_46_re) - t_1)) * sin(((t_0 * y_46_im) + t_2));
	} else if (x_46_im <= 3.6e-166) {
		tmp = exp(((log(x_46_re) * y_46_re) - t_1)) * sin(((log(x_46_re) * y_46_im) + t_2));
	} else if (x_46_im <= 1.4e-124) {
		tmp = t_4 * sin(((log(-x_46_re) * y_46_im) + t_2));
	} else if (x_46_im <= 1.16e+14) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_3;
	} else {
		tmp = t_4 * t_3;
	}
	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 = log(-x_46im)
    t_1 = atan2(x_46im, x_46re) * y_46im
    t_2 = atan2(x_46im, x_46re) * y_46re
    t_3 = sin(((log(x_46im) * y_46im) + t_2))
    t_4 = exp(((log(x_46im) * y_46re) - t_1))
    if (x_46im <= (-2.25d-299)) then
        tmp = exp(((t_0 * y_46re) - t_1)) * sin(((t_0 * y_46im) + t_2))
    else if (x_46im <= 3.6d-166) then
        tmp = exp(((log(x_46re) * y_46re) - t_1)) * sin(((log(x_46re) * y_46im) + t_2))
    else if (x_46im <= 1.4d-124) then
        tmp = t_4 * sin(((log(-x_46re) * y_46im) + t_2))
    else if (x_46im <= 1.16d+14) then
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - t_1)) * t_3
    else
        tmp = t_4 * t_3
    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 = Math.log(-x_46_im);
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_3 = Math.sin(((Math.log(x_46_im) * y_46_im) + t_2));
	double t_4 = Math.exp(((Math.log(x_46_im) * y_46_re) - t_1));
	double tmp;
	if (x_46_im <= -2.25e-299) {
		tmp = Math.exp(((t_0 * y_46_re) - t_1)) * Math.sin(((t_0 * y_46_im) + t_2));
	} else if (x_46_im <= 3.6e-166) {
		tmp = Math.exp(((Math.log(x_46_re) * y_46_re) - t_1)) * Math.sin(((Math.log(x_46_re) * y_46_im) + t_2));
	} else if (x_46_im <= 1.4e-124) {
		tmp = t_4 * Math.sin(((Math.log(-x_46_re) * y_46_im) + t_2));
	} else if (x_46_im <= 1.16e+14) {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_3;
	} else {
		tmp = t_4 * t_3;
	}
	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 = math.log(-x_46_im)
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_3 = math.sin(((math.log(x_46_im) * y_46_im) + t_2))
	t_4 = math.exp(((math.log(x_46_im) * y_46_re) - t_1))
	tmp = 0
	if x_46_im <= -2.25e-299:
		tmp = math.exp(((t_0 * y_46_re) - t_1)) * math.sin(((t_0 * y_46_im) + t_2))
	elif x_46_im <= 3.6e-166:
		tmp = math.exp(((math.log(x_46_re) * y_46_re) - t_1)) * math.sin(((math.log(x_46_re) * y_46_im) + t_2))
	elif x_46_im <= 1.4e-124:
		tmp = t_4 * math.sin(((math.log(-x_46_re) * y_46_im) + t_2))
	elif x_46_im <= 1.16e+14:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_3
	else:
		tmp = t_4 * t_3
	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 = log(Float64(-x_46_im))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_3 = sin(Float64(Float64(log(x_46_im) * y_46_im) + t_2))
	t_4 = exp(Float64(Float64(log(x_46_im) * y_46_re) - t_1))
	tmp = 0.0
	if (x_46_im <= -2.25e-299)
		tmp = Float64(exp(Float64(Float64(t_0 * y_46_re) - t_1)) * sin(Float64(Float64(t_0 * y_46_im) + t_2)));
	elseif (x_46_im <= 3.6e-166)
		tmp = Float64(exp(Float64(Float64(log(x_46_re) * y_46_re) - t_1)) * sin(Float64(Float64(log(x_46_re) * y_46_im) + t_2)));
	elseif (x_46_im <= 1.4e-124)
		tmp = Float64(t_4 * sin(Float64(Float64(log(Float64(-x_46_re)) * y_46_im) + t_2)));
	elseif (x_46_im <= 1.16e+14)
		tmp = 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_1)) * t_3);
	else
		tmp = Float64(t_4 * t_3);
	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 = log(-x_46_im);
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	t_2 = atan2(x_46_im, x_46_re) * y_46_re;
	t_3 = sin(((log(x_46_im) * y_46_im) + t_2));
	t_4 = exp(((log(x_46_im) * y_46_re) - t_1));
	tmp = 0.0;
	if (x_46_im <= -2.25e-299)
		tmp = exp(((t_0 * y_46_re) - t_1)) * sin(((t_0 * y_46_im) + t_2));
	elseif (x_46_im <= 3.6e-166)
		tmp = exp(((log(x_46_re) * y_46_re) - t_1)) * sin(((log(x_46_re) * y_46_im) + t_2));
	elseif (x_46_im <= 1.4e-124)
		tmp = t_4 * sin(((log(-x_46_re) * y_46_im) + t_2));
	elseif (x_46_im <= 1.16e+14)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * t_3;
	else
		tmp = t_4 * t_3;
	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[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -2.25e-299], N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3.6e-166], N[(N[Exp[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.4e-124], N[(t$95$4 * N[Sin[N[(N[(N[Log[(-x$46$re)], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.16e+14], 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$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(t$95$4 * t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.im < -2.25000000000000001e-299

   1. Initial program 34.0

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

   2. Taylor expanded in x.im around -inf 24.1

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

   3. Simplified 24.1

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

      [Start] 24.1	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-2 [=>] 24.1	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(x.im \cdot -1\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-9 [=>] 24.1	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.im\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. Taylor expanded in x.im around -inf 17.1

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

   5. Simplified 17.1

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

      [Start] 17.1	\[ e^{\log \left(-1 \cdot x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-2 [=>] 17.1	\[ e^{\log \color{blue}{\left(x.im \cdot -1\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-9 [=>] 17.1	\[ e^{\log \color{blue}{\left(-x.im\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if -2.25000000000000001e-299 < x.im < 3.6000000000000001e-166

   1. Initial program 34.6

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

   2. Taylor expanded in x.re around inf 44.1

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

   3. Taylor expanded in x.re around inf 37.3

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

   if 3.6000000000000001e-166 < x.im < 1.39999999999999999e-124

   1. Initial program 21.4

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

   2. Taylor expanded in x.re around 0 27.1

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

   3. Taylor expanded in x.re around -inf 40.7

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

   4. Simplified 40.7

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

      [Start] 40.7	\[ e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-2 [=>] 40.7	\[ e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(x.re \cdot -1\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      rational.json-simplify-9 [=>] 40.7	\[ e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if 1.39999999999999999e-124 < x.im < 1.16e14

   1. Initial program 17.2

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

   2. Taylor expanded in x.re around 0 15.6

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

   if 1.16e14 < x.im

   1. Initial program 42.1

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

   2. Taylor expanded in x.re around 0 42.1

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

   3. Taylor expanded in x.re around 0 4.7

      \[\leadsto e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{x.im} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.25 \cdot 10^{-299}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 1.16 \cdot 10^{+14}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.im \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.im \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	53320

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \log \left(-x.im\right)\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -3 \cdot 10^{-307}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot \sin \left(t_1 \cdot y.im + t_0\right)\\ \mathbf{elif}\;x.im \leq 1.16 \cdot 10^{+14}:\\ \;\;\;\;e^{t_2 \cdot y.re - t_3} \cdot \sin \left(t_2 \cdot y.im + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_3} \cdot \sin \left(\log x.im \cdot y.im + t_0\right)\\ \end{array} \]

Alternative 2
Error	17.9
Cost	46800

\[\begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_4 := e^{\log x.im \cdot y.re - t_2}\\ \mathbf{if}\;x.im \leq -2.36 \cdot 10^{-299}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_2} \cdot \sin \left(t_1 \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.im \leq 1.02 \cdot 10^{-165}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_2} \cdot \sin \left(\log x.re \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.im \leq 6.2 \cdot 10^{-147}:\\ \;\;\;\;t_4 \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.im \leq 3.2 \cdot 10^{-82}:\\ \;\;\;\;t_4 \cdot \sin \left(\log \left(\sqrt{t_0}\right) \cdot y.im + t_3\right)\\ \mathbf{elif}\;x.im \leq 10^{-11}:\\ \;\;\;\;e^{\frac{y.re \cdot \log t_0}{2} - t_2} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \sin \left(\log x.im \cdot y.im + t_3\right)\\ \end{array} \]

Alternative 3
Error	17.9
Cost	40016

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\log x.re \cdot y.re - t_1} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_3 := e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - t_1} \cdot t_0\\ \mathbf{if}\;x.re \leq -1:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t_1} \cdot t_0\\ \mathbf{elif}\;x.re \leq 7.8 \cdot 10^{-306}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.re \leq 1.12 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	18.3
Cost	39948

\[\begin{array}{l} t_0 := \log \left(-x.im\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := e^{\log x.im \cdot y.re - t_1}\\ \mathbf{if}\;x.im \leq -1.9 \cdot 10^{-299}:\\ \;\;\;\;e^{t_0 \cdot y.re - t_1} \cdot \sin \left(t_0 \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{-170}:\\ \;\;\;\;e^{\log x.re \cdot y.re - t_1} \cdot \sin \left(\log x.re \cdot y.im + t_2\right)\\ \mathbf{elif}\;x.im \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;t_3 \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin \left(\log x.im \cdot y.im + t_2\right)\\ \end{array} \]

Alternative 5
Error	20.3
Cost	33552

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - t_0} \cdot t_1\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -6.7 \cdot 10^{-224}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_0} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.re \leq 380000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log \left({x.im}^{2}\right) \cdot y.re}{2} - t_0} \cdot t_1\\ \end{array} \]

Alternative 6
Error	20.7
Cost	33164

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - t_0} \cdot t_1\\ t_3 := t_1 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-192}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-224}:\\ \;\;\;\;e^{\log x.im \cdot y.re - t_0} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.re \leq 29000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	20.4
Cost	27208

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\frac{y.re \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}{2} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t_0\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 540000000:\\ \;\;\;\;t_0 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	24.2
Cost	26828

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := e^{\log x.re \cdot y.re - t_1} \cdot t_0\\ \mathbf{if}\;x.re \leq -2.95 \cdot 10^{-307}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - t_1} \cdot t_0\\ \mathbf{elif}\;x.re \leq 1.45 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-61}:\\ \;\;\;\;\left(\left(\sin t_0 - -1\right) - 1\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	27.4
Cost	20104

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -62000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{+97}:\\ \;\;\;\;t_0 \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	37.3
Cost	13512

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -430:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 7.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	51.1
Cost	6656

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

Reproduce

herbie shell --seed 2023073 
(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)))))