raw-angle from scale-rotated-ellipse

Average Error: 55.2 → 27.7

Time: 2.4min

Precision: binary64

Cost: 46344

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_2 := \cos t_0\\ t_3 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_4 := -\cos t_3\\ t_5 := \sin t_1\\ t_6 := \sin t_3\\ \mathbf{if}\;b \leq -4.7 \cdot 10^{+144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{t_2}{\sin t_0}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -8.8 \cdot 10^{+104}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|t_5\right|}{\cos t_1}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.28 \cdot 10^{-27}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{t_4}{t_6}\right)}{\pi}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+33}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t_5}{t_2}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_4}{x-scale \cdot t_6}\right)}{\pi}\\ \end{array} \]

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 (* angle (* PI 0.005555555555555556)))
        (t_2 (cos t_0))
        (t_3 (* 0.005555555555555556 (* PI angle)))
        (t_4 (- (cos t_3)))
        (t_5 (sin t_1))
        (t_6 (sin t_3)))
   (if (<= b -4.7e+144)
     (*
      180.0
      (/ (atan (* -0.5 (* (/ y-scale x-scale) (* 2.0 (/ t_2 (sin t_0)))))) PI))
     (if (<= b -8.8e+104)
       (*
        180.0
        (/
         (atan
          (* -0.5 (* (/ y-scale x-scale) (/ 2.0 (/ (fabs t_5) (cos t_1))))))
         PI))
       (if (<= b -1.28e-27)
         (* 180.0 (/ (atan (* (/ y-scale x-scale) (/ t_4 t_6))) PI))
         (if (<= b 8e+33)
           (*
            180.0
            (/
             (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_5 t_2)))))
             PI))
           (* 180.0 (/ (atan (/ (* y-scale t_4) (* x-scale t_6))) 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 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_2 = cos(t_0);
	double t_3 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_4 = -cos(t_3);
	double t_5 = sin(t_1);
	double t_6 = sin(t_3);
	double tmp;
	if (b <= -4.7e+144) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (t_2 / sin(t_0)))))) / ((double) M_PI));
	} else if (b <= -8.8e+104) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 / (fabs(t_5) / cos(t_1)))))) / ((double) M_PI));
	} else if (b <= -1.28e-27) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_4 / t_6))) / ((double) M_PI));
	} else if (b <= 8e+33) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_5 / t_2))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * t_4) / (x_45_scale * t_6))) / ((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 = angle * (Math.PI * 0.005555555555555556);
	double t_2 = Math.cos(t_0);
	double t_3 = 0.005555555555555556 * (Math.PI * angle);
	double t_4 = -Math.cos(t_3);
	double t_5 = Math.sin(t_1);
	double t_6 = Math.sin(t_3);
	double tmp;
	if (b <= -4.7e+144) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (t_2 / Math.sin(t_0)))))) / Math.PI);
	} else if (b <= -8.8e+104) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 / (Math.abs(t_5) / Math.cos(t_1)))))) / Math.PI);
	} else if (b <= -1.28e-27) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * (t_4 / t_6))) / Math.PI);
	} else if (b <= 8e+33) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_5 / t_2))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_4) / (x_45_scale * t_6))) / 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 = angle * (math.pi * 0.005555555555555556)
	t_2 = math.cos(t_0)
	t_3 = 0.005555555555555556 * (math.pi * angle)
	t_4 = -math.cos(t_3)
	t_5 = math.sin(t_1)
	t_6 = math.sin(t_3)
	tmp = 0
	if b <= -4.7e+144:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (t_2 / math.sin(t_0)))))) / math.pi)
	elif b <= -8.8e+104:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 / (math.fabs(t_5) / math.cos(t_1)))))) / math.pi)
	elif b <= -1.28e-27:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * (t_4 / t_6))) / math.pi)
	elif b <= 8e+33:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_5 / t_2))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * t_4) / (x_45_scale * t_6))) / 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(angle * Float64(pi * 0.005555555555555556))
	t_2 = cos(t_0)
	t_3 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_4 = Float64(-cos(t_3))
	t_5 = sin(t_1)
	t_6 = sin(t_3)
	tmp = 0.0
	if (b <= -4.7e+144)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(2.0 * Float64(t_2 / sin(t_0)))))) / pi));
	elseif (b <= -8.8e+104)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(2.0 / Float64(abs(t_5) / cos(t_1)))))) / pi));
	elseif (b <= -1.28e-27)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(t_4 / t_6))) / pi));
	elseif (b <= 8e+33)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_5 / t_2))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_4) / Float64(x_45_scale * t_6))) / 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 = angle * (pi * 0.005555555555555556);
	t_2 = cos(t_0);
	t_3 = 0.005555555555555556 * (pi * angle);
	t_4 = -cos(t_3);
	t_5 = sin(t_1);
	t_6 = sin(t_3);
	tmp = 0.0;
	if (b <= -4.7e+144)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (t_2 / sin(t_0)))))) / pi);
	elseif (b <= -8.8e+104)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 / (abs(t_5) / cos(t_1)))))) / pi);
	elseif (b <= -1.28e-27)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_4 / t_6))) / pi);
	elseif (b <= 8e+33)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_5 / t_2))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * t_4) / (x_45_scale * t_6))) / 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[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-N[Cos[t$95$3], $MachinePrecision])}, Block[{t$95$5 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[b, -4.7e+144], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(2.0 * N[(t$95$2 / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.8e+104], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(2.0 / N[(N[Abs[t$95$5], $MachinePrecision] / N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.28e-27], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$4 / t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e+33], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$5 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$4), $MachinePrecision] / N[(x$45$scale * t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4.7000000000000002e144

   1. Initial program 63.1

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

   2. Simplified 63.0

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

      [Start] 63.1

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

   3. Taylor expanded in x-scale around 0 62.2

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

   4. Simplified 61.8

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

      [Start] 62.2

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

      times-frac [=>] 61.7

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

   5. Taylor expanded in b around inf 26.7

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

   6. Simplified 26.8

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

      [Start] 26.7	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*r* [=>] 26.7	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      *-commutative [<=] 26.7	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*r* [=>] 26.8	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]
      *-commutative [<=] 26.8	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}\right)\right)\right)}{\pi} \]

   if -4.7000000000000002e144 < b < -8.80000000000000002e104

   1. Initial program 49.4

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

   2. Simplified 51.0

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

      [Start] 49.4

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

   3. Taylor expanded in x-scale around 0 36.8

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

   4. Simplified 34.8

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

      [Start] 36.8

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

      times-frac [=>] 33.0

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

   5. Applied egg-rr 43.5

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

   6. Taylor expanded in b around inf 36.9

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

   7. Simplified 36.0

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

      [Start] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\left|\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|}\right)\right)\right)}{\pi} \]
      associate-*r/ [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\frac{2 \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\left|\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|}}\right)\right)}{\pi} \]
      associate-/l* [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\frac{2}{\frac{\left|\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}}\right)\right)}{\pi} \]
      associate-*r* [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}{\pi} \]
      *-commutative [<=] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}{\pi} \]
      associate-*r* [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}{\pi} \]
      *-commutative [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}{\pi} \]
      *-commutative [=>] 36.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right)\right|}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}{\pi} \]
      associate-*r* [=>] 38.0	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right|}{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}}\right)\right)}{\pi} \]
      *-commutative [<=] 38.0	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right|}{\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right)}{\pi} \]
      associate-*r* [=>] 36.0	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right|}{\cos \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}}}\right)\right)}{\pi} \]
      *-commutative [=>] 36.0	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right|}{\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}}\right)\right)}{\pi} \]
      *-commutative [=>] 36.0	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right|}{\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right)}}\right)\right)}{\pi} \]

   if -8.80000000000000002e104 < b < -1.27999999999999993e-27

   1. Initial program 50.2

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

   2. Simplified 48.4

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

      [Start] 50.2

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

   3. Taylor expanded in y-scale around inf 44.2

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

   4. Simplified 43.9

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

      [Start] 44.2

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

   5. Taylor expanded in b around inf 37.0

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

   6. Simplified 35.9

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

      [Start] 37.0	\[ 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
      mul-1-neg [=>] 37.0	\[ 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
      times-frac [=>] 35.9	\[ 180 \cdot \frac{\tan^{-1} \left(-\color{blue}{\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if -1.27999999999999993e-27 < b < 7.9999999999999996e33

   1. Initial program 53.5

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

   2. Simplified 54.0

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

      [Start] 53.5

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

   3. Taylor expanded in x-scale around 0 44.0

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

   4. Simplified 42.0

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

      [Start] 44.0

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

      times-frac [=>] 42.3

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

   5. Taylor expanded in b around 0 25.8

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

   6. Simplified 25.4

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

      [Start] 25.8	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*r* [=>] 25.8	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      *-commutative [<=] 25.8	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*r* [=>] 25.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]
      *-commutative [<=] 25.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}\right)\right)\right)}{\pi} \]

   7. Taylor expanded in angle around 0 25.9

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

   8. Simplified 25.5

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

      [Start] 25.9	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*r* [=>] 25.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi} \]
      *-commutative [=>] 25.4	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi} \]
      associate-*l* [=>] 25.5	\[ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi} \]

   if 7.9999999999999996e33 < b

   1. Initial program 57.3

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

   2. Simplified 56.6

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

      [Start] 57.3

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

   3. Taylor expanded in y-scale around inf 54.8

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

   4. Simplified 54.7

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

      [Start] 54.8

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

   5. Taylor expanded in b around inf 28.2

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -8.8 \cdot 10^{+104}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{2}{\frac{\left|\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right|}{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.28 \cdot 10^{-27}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{-\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+33}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi}\\ \end{array} \]

Alternatives

Alternative 1
Error	34.2
Cost	40544

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{-\cos t_2}{\sin t_2}\right)}{\pi}\\ \mathbf{if}\;x-scale \leq -3 \cdot 10^{-45}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{t_1}{\cos t_0}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq -1.4 \cdot 10^{-185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq -1.3 \cdot 10^{-215}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale}{\mathsf{fma}\left(-90, \frac{\frac{x-scale}{angle}}{\pi}, \left(x-scale \cdot \left(\pi \cdot -0.001851851851851852\right)\right) \cdot \left(-0.5 \cdot angle\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq -5.2 \cdot 10^{-249}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\log \left(e^{\pi \cdot \left(y-scale \cdot angle\right)}\right)}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 8.5 \cdot 10^{-305}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(t_1 \cdot -2\right)\right)\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 5.5 \cdot 10^{-253}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 5.6 \cdot 10^{-167}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(\frac{y-scale}{x-scale} \cdot \tan t_0\right)\right)\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 6.8 \cdot 10^{-49}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(y-scale \cdot -2\right)}{x-scale}\right)}{\pi}\\ \end{array} \]

Alternative 2
Error	28.4
Cost	40209

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t_1}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)}{\pi}\\ t_3 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq -6 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-181}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-\cos t_3\right)}{x-scale \cdot \sin t_3}\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-56} \lor \neg \left(a \leq 1.05 \cdot 10^{+16}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\cos t_0}{t_1} \cdot \frac{-y-scale}{x-scale}\right)}{\pi}\\ \end{array} \]

Alternative 3
Error	27.6
Cost	39753

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{-33} \lor \neg \left(b \leq 1.3 \cdot 10^{+33}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-\cos t_0\right)}{x-scale \cdot \sin t_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{-2 \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\frac{x-scale}{y-scale}}\right)}{\pi}\\ \end{array} \]

Alternative 4
Error	27.5
Cost	39752

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;b \leq -9.5 \cdot 10^{-28}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{-\cos t_0}{\sin t_0}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+32}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{-2 \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\frac{x-scale}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\cos t_1}{\sin t_1} \cdot \frac{-y-scale}{x-scale}\right)}{\pi}\\ \end{array} \]

Alternative 5
Error	29.6
Cost	27352

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{y-scale \cdot -2}{x-scale \cdot \left(\pi \cdot angle\right)}\right)}{\pi}\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot -2\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-29}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{angle} \cdot \frac{-2}{x-scale \cdot \pi}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+32}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{-2 \cdot \tan t_1}{\frac{x-scale}{y-scale}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{b \cdot b}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+196}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \sin t_1\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	29.2
Cost	27088

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{y-scale \cdot -2}{x-scale \cdot \left(\pi \cdot angle\right)}\right)}{\pi}\\ t_1 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{-2 \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\frac{x-scale}{y-scale}}\right)}{\pi}\\ \mathbf{if}\;b \leq -8 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.75 \cdot 10^{-30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{angle} \cdot \frac{-2}{x-scale \cdot \pi}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.95 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	29.4
Cost	27088

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{y-scale \cdot -2}{x-scale \cdot \left(\pi \cdot angle\right)}\right)}{\pi}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4 \cdot 10^{+28}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot -2\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.42 \cdot 10^{-27}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{angle} \cdot \frac{-2}{x-scale \cdot \pi}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+32}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{-2 \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\frac{x-scale}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	29.5
Cost	26825

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-28} \lor \neg \left(b \leq 5.2 \cdot 10^{+33}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{y-scale \cdot -2}{x-scale \cdot \left(\pi \cdot angle\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \left(\frac{y-scale}{x-scale} \cdot \tan \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \]

Alternative 9
Error	33.5
Cost	20560

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot angle}{\frac{x-scale}{\pi}}\right)\right)}{\pi}\\ \mathbf{if}\;a \leq -2.05 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-192}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{angle} \cdot \frac{-2}{x-scale \cdot \pi}\right)\right)}{\pi}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-29}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale}{-90 \cdot \frac{\frac{x-scale}{angle}}{\pi}}\right)}{\pi}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{\frac{y-scale \cdot -2}{x-scale}}{\pi \cdot angle}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	33.4
Cost	20297

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-56} \lor \neg \left(b \leq 2.1 \cdot 10^{+204}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{angle} \cdot \frac{-2}{x-scale \cdot \pi}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot angle}{\frac{x-scale}{\pi}}\right)\right)}{\pi}\\ \end{array} \]

Alternative 11
Error	33.3
Cost	20297

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-59} \lor \neg \left(b \leq 1.75 \cdot 10^{+196}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{y-scale \cdot -2}{x-scale \cdot \left(\pi \cdot angle\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot angle}{\frac{x-scale}{\pi}}\right)\right)}{\pi}\\ \end{array} \]

Alternative 12
Error	37.9
Cost	20032

\[180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot angle}{\frac{x-scale}{\pi}}\right)\right)}{\pi} \]

Alternative 13
Error	56.4
Cost	19904

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

Alternative 14
Error	56.4
Cost	19904

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

Reproduce

herbie shell --seed 2023027 
(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)))