raw-angle from scale-rotated-ellipse

Average Error: 54.9 → 28.3

Time: 1.8min

Precision: binary64

Cost: 39812

Math

\[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t_1\\ \mathbf{if}\;a \leq -4 \cdot 10^{+24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin t_0}{\cos t_0 \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot t_2}{x-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_1}{x-scale \cdot t_2}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/
   (atan
    (/
     (-
      (-
       (/
        (/
         (+
          (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
         y-scale)
        y-scale)
       (/
        (/
         (+
          (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
         x-scale)
        x-scale))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
            x-scale)
           x-scale)
          (/
           (/
            (+
             (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
            y-scale)
           y-scale))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 2.0 (- (pow b 2.0) (pow a 2.0)))
             (sin (* (/ angle 180.0) PI)))
            (cos (* (/ angle 180.0) PI)))
           x-scale)
          y-scale)
         2.0))))
     (/
      (/
       (*
        (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
        (cos (* (/ angle 180.0) PI)))
       x-scale)
      y-scale)))
   PI)))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1)))
   (if (<= a -4e+24)
     (*
      180.0
      (/
       (atan (* -0.5 (* -2.0 (/ (* y-scale (sin t_0)) (* (cos t_0) x-scale)))))
       PI))
     (if (<= a 1.8e+16)
       (*
        180.0
        (/
         (atan
          (-
           (/
            (* y-scale t_2)
            (* x-scale (sin (* PI (* angle 0.005555555555555556)))))))
         PI))
       (* 180.0 (/ (atan (/ (* y-scale (sin t_1)) (* x-scale t_2))) PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(((((((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale) - (((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale)) - sqrt((pow(((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) - (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale), 2.0)))) / (((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale))) / ((double) M_PI));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double tmp;
	if (a <= -4e+24) {
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * sin(t_0)) / (cos(t_0) * x_45_scale))))) / ((double) M_PI));
	} else if (a <= 1.8e+16) {
		tmp = 180.0 * (atan(-((y_45_scale * t_2) / (x_45_scale * sin((((double) M_PI) * (angle * 0.005555555555555556)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_1)) / (x_45_scale * t_2))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(((((((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale) - (((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale)) - Math.sqrt((Math.pow(((((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale) - (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale), 2.0)))) / (((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale))) / Math.PI);
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.cos(t_1);
	double tmp;
	if (a <= -4e+24) {
		tmp = 180.0 * (Math.atan((-0.5 * (-2.0 * ((y_45_scale * Math.sin(t_0)) / (Math.cos(t_0) * x_45_scale))))) / Math.PI);
	} else if (a <= 1.8e+16) {
		tmp = 180.0 * (Math.atan(-((y_45_scale * t_2) / (x_45_scale * Math.sin((Math.PI * (angle * 0.005555555555555556)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_1)) / (x_45_scale * t_2))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(((((((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale) - (((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale)) - math.sqrt((math.pow(((((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale) - (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale), 2.0)))) / (((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale))) / math.pi)

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.cos(t_1)
	tmp = 0
	if a <= -4e+24:
		tmp = 180.0 * (math.atan((-0.5 * (-2.0 * ((y_45_scale * math.sin(t_0)) / (math.cos(t_0) * x_45_scale))))) / math.pi)
	elif a <= 1.8e+16:
		tmp = 180.0 * (math.atan(-((y_45_scale * t_2) / (x_45_scale * math.sin((math.pi * (angle * 0.005555555555555556)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin(t_1)) / (x_45_scale * t_2))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale) - Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale)) - sqrt(Float64((Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0)))) / Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale))) / pi))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	tmp = 0.0
	if (a <= -4e+24)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * sin(t_0)) / Float64(cos(t_0) * x_45_scale))))) / pi));
	elseif (a <= 1.8e+16)
		tmp = Float64(180.0 * Float64(atan(Float64(-Float64(Float64(y_45_scale * t_2) / Float64(x_45_scale * sin(Float64(pi * Float64(angle * 0.005555555555555556))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_1)) / Float64(x_45_scale * t_2))) / pi));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(((((((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale) - (((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale)) - sqrt(((((((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0)))) / (((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale))) / pi);
end

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = cos(t_1);
	tmp = 0.0;
	if (a <= -4e+24)
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * sin(t_0)) / (cos(t_0) * x_45_scale))))) / pi);
	elseif (a <= 1.8e+16)
		tmp = 180.0 * (atan(-((y_45_scale * t_2) / (x_45_scale * sin((pi * (angle * 0.005555555555555556)))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * sin(t_1)) / (x_45_scale * t_2))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] - N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] - N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[a, -4e+24], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+16], N[(180.0 * N[(N[ArcTan[(-N[(N[(y$45$scale * t$95$2), $MachinePrecision] / N[(x$45$scale * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.9999999999999999e24

   1. Initial program 59.6

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Simplified 59.6

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}} \]
      Proof

      [Start] 59.6

      \[ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   3. Taylor expanded in x-scale around 0 52.8

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}}{\pi} \]

   4. Taylor expanded in b around 0 27.2

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \color{blue}{\left(-2 \cdot \frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}\right)}{\pi} \]

   5. Simplified 27.2

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \color{blue}{\left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale}\right)}\right)}{\pi} \]
      Proof

      [Start] 27.2	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.2	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot x-scale}}\right)\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot x-scale}\right)\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.2	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot x-scale}\right)\right)}{\pi} \]

   if -3.9999999999999999e24 < a < 1.8e16

   1. Initial program 50.9

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Simplified 50.9

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}} \]
      Proof

      [Start] 50.9

      \[ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   3. Taylor expanded in x-scale around 0 48.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   4. Simplified 48.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{2 \cdot \left({\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot {a}^{2}\right) + {\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot \left({b}^{2} + {b}^{2}\right)}{{x-scale}^{2}}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
      Proof

      [Start] 48.3

      \[ 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   5. Taylor expanded in angle around inf 48.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \frac{2 \cdot \left({\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{2} \cdot {a}^{2}\right) + {\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot \left({b}^{2} + {b}^{2}\right)}{{x-scale}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   6. Taylor expanded in a around 0 28.8

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   7. Simplified 28.8

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
      Proof

      [Start] 28.8	\[ 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 28.8	\[ 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot -1\right)}}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 28.8	\[ 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   8. Applied egg-rr 28.7

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right) + 0\right)}}\right)}{\pi} \]

   9. Simplified 28.7

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)}{\pi} \]
      Proof

      [Start] 28.7	\[ 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right) + 0\right)}\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 28.7	\[ 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}\right)}{\pi} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 28.7	\[ 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}\right)}{\pi} \]

   if 1.8e16 < a

   1. Initial program 59.3

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Simplified 59.3

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}} \]
      Proof

      [Start] 59.3

      \[ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   3. Taylor expanded in x-scale around 0 55.5

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   4. Simplified 55.5

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{2 \cdot \left({\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot {a}^{2}\right) + {\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2} \cdot \left({b}^{2} + {b}^{2}\right)}{{x-scale}^{2}}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
      Proof

      [Start] 55.5

      \[ 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   5. Taylor expanded in a around inf 28.6

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]

Alternatives

Alternative 1
Error	28.7
Cost	39952

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t_0\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_1}{x-scale \cdot \cos t_0}\right)}{\pi}\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot 1}{x-scale \cdot t_1}\right)}{\pi}\\ \mathbf{if}\;a \leq -2.16 \cdot 10^{+27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-89}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-70}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{1 \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	28.6
Cost	39952

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t_1\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot 1}{x-scale \cdot t_2}\right)}{\pi}\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_0}{x-scale \cdot \cos t_0}\right)}{\pi}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-71}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{1 \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_2}{x-scale \cdot \cos t_1}\right)}{\pi}\\ \end{array} \]

Alternative 3
Error	28.5
Cost	39752

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq -2.25 \cdot 10^{+23}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_1}{x-scale \cdot t_2}\right)}{\pi}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale \cdot \left(t_1 \cdot t_2\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_3}{x-scale \cdot \cos t_3}\right)}{\pi}\\ \end{array} \]

Alternative 4
Error	28.3
Cost	39752

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t_1\\ t_3 := \sin t_1\\ \mathbf{if}\;a \leq -6 \cdot 10^{+19}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_0}{x-scale \cdot \cos t_0}\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot t_2}{x-scale \cdot t_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_3}{x-scale \cdot t_2}\right)}{\pi}\\ \end{array} \]

Alternative 5
Error	28.3
Cost	39752

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t_1\\ \mathbf{if}\;a \leq -5 \cdot 10^{+22}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_0}{x-scale \cdot \cos t_0}\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot t_2}{x-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t_1}{x-scale \cdot t_2}\right)}{\pi}\\ \end{array} \]

Alternative 6
Error	29.5
Cost	33412

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot 1}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin t_1}{1 \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{+56}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{\cos t_1 \cdot x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	29.5
Cost	27216

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{1 \cdot x-scale}\right)\right)}{\pi}\\ t_1 := 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot 1}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(0.005555555555555556 \cdot \frac{angle \cdot \left(\pi \cdot y-scale\right)}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	30.5
Cost	26760

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot 1}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{if}\;b \leq -90000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-35}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(0.005555555555555556 \cdot \frac{y-scale \cdot \left(angle \cdot \pi\right)}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	32.3
Cost	20424

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{x-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-32}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(0.005555555555555556 \cdot \frac{angle \cdot \left(\pi \cdot y-scale\right)}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	32.3
Cost	20424

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{x-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-32}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(0.005555555555555556 \cdot \frac{y-scale \cdot \left(angle \cdot \pi\right)}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	56.1
Cost	19904

\[180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{y-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi} \]

Alternative 12
Error	39.7
Cost	19904

\[180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{x-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (a b angle x-scale y-scale)
  :name "raw-angle from scale-rotated-ellipse"
  :precision binary64
  (* 180.0 (/ (atan (/ (- (- (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale) (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0)))) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale))) PI)))