raw-angle from scale-rotated-ellipse

Average Error: 54.8 → 30.4

Time: 1.6min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t_1\\ t_3 := \sin t_0\\ t_4 := 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{{\cos t_0}^{2}}{x-scale \cdot x-scale} \cdot \frac{-y-scale}{t_3}\right)\right)}{\pi}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{+120}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{+88}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t_2}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{y-scale \cdot \left(-{\cos t_1}^{2}\right)}{t_2 \cdot {x-scale}^{2}}\right)}{\pi}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-43}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(0.005555555555555556 \cdot \left(\frac{\pi}{x-scale} \cdot \frac{angle}{\frac{x-scale}{y-scale}}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-237}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t_3 \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t_0 \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (* angle (* PI 0.005555555555555556)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_1))
        (t_3 (sin t_0))
        (t_4
         (*
          180.0
          (/
           (atan
            (*
             x-scale
             (*
              (/ (pow (cos t_0) 2.0) (* x-scale x-scale))
              (/ (- y-scale) t_3))))
           PI))))
   (if (<= b -2.1e+120)
     t_4
     (if (<= b -4.4e+88)
       (* 180.0 (/ (atan (/ (* y-scale t_2) x-scale)) PI))
       (if (<= b -1.75e-6)
         (*
          180.0
          (/
           (atan
            (*
             x-scale
             (/
              (* y-scale (- (pow (cos t_1) 2.0)))
              (* t_2 (pow x-scale 2.0)))))
           PI))
         (if (<= b -6.4e-43)
           (*
            180.0
            (/
             (atan
              (*
               x-scale
               (*
                0.005555555555555556
                (* (/ PI x-scale) (/ angle (/ x-scale y-scale))))))
             PI))
           (if (<= b -1.7e-82)
             t_4
             (if (<= b 3e-237)
               (* 180.0 (/ (atan (* t_3 (/ y-scale x-scale))) PI))
               (if (<= b 1.25e+39)
                 (* 180.0 (/ (atan (* t_0 (/ y-scale x-scale))) PI))
                 t_4)))))))))

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 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_1);
	double t_3 = sin(t_0);
	double t_4 = 180.0 * (atan((x_45_scale * ((pow(cos(t_0), 2.0) / (x_45_scale * x_45_scale)) * (-y_45_scale / t_3)))) / ((double) M_PI));
	double tmp;
	if (b <= -2.1e+120) {
		tmp = t_4;
	} else if (b <= -4.4e+88) {
		tmp = 180.0 * (atan(((y_45_scale * t_2) / x_45_scale)) / ((double) M_PI));
	} else if (b <= -1.75e-6) {
		tmp = 180.0 * (atan((x_45_scale * ((y_45_scale * -pow(cos(t_1), 2.0)) / (t_2 * pow(x_45_scale, 2.0))))) / ((double) M_PI));
	} else if (b <= -6.4e-43) {
		tmp = 180.0 * (atan((x_45_scale * (0.005555555555555556 * ((((double) M_PI) / x_45_scale) * (angle / (x_45_scale / y_45_scale)))))) / ((double) M_PI));
	} else if (b <= -1.7e-82) {
		tmp = t_4;
	} else if (b <= 3e-237) {
		tmp = 180.0 * (atan((t_3 * (y_45_scale / x_45_scale))) / ((double) M_PI));
	} else if (b <= 1.25e+39) {
		tmp = 180.0 * (atan((t_0 * (y_45_scale / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = t_4;
	}
	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 = angle * (Math.PI * 0.005555555555555556);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.sin(t_1);
	double t_3 = Math.sin(t_0);
	double t_4 = 180.0 * (Math.atan((x_45_scale * ((Math.pow(Math.cos(t_0), 2.0) / (x_45_scale * x_45_scale)) * (-y_45_scale / t_3)))) / Math.PI);
	double tmp;
	if (b <= -2.1e+120) {
		tmp = t_4;
	} else if (b <= -4.4e+88) {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_2) / x_45_scale)) / Math.PI);
	} else if (b <= -1.75e-6) {
		tmp = 180.0 * (Math.atan((x_45_scale * ((y_45_scale * -Math.pow(Math.cos(t_1), 2.0)) / (t_2 * Math.pow(x_45_scale, 2.0))))) / Math.PI);
	} else if (b <= -6.4e-43) {
		tmp = 180.0 * (Math.atan((x_45_scale * (0.005555555555555556 * ((Math.PI / x_45_scale) * (angle / (x_45_scale / y_45_scale)))))) / Math.PI);
	} else if (b <= -1.7e-82) {
		tmp = t_4;
	} else if (b <= 3e-237) {
		tmp = 180.0 * (Math.atan((t_3 * (y_45_scale / x_45_scale))) / Math.PI);
	} else if (b <= 1.25e+39) {
		tmp = 180.0 * (Math.atan((t_0 * (y_45_scale / x_45_scale))) / Math.PI);
	} else {
		tmp = t_4;
	}
	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 = angle * (math.pi * 0.005555555555555556)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.sin(t_1)
	t_3 = math.sin(t_0)
	t_4 = 180.0 * (math.atan((x_45_scale * ((math.pow(math.cos(t_0), 2.0) / (x_45_scale * x_45_scale)) * (-y_45_scale / t_3)))) / math.pi)
	tmp = 0
	if b <= -2.1e+120:
		tmp = t_4
	elif b <= -4.4e+88:
		tmp = 180.0 * (math.atan(((y_45_scale * t_2) / x_45_scale)) / math.pi)
	elif b <= -1.75e-6:
		tmp = 180.0 * (math.atan((x_45_scale * ((y_45_scale * -math.pow(math.cos(t_1), 2.0)) / (t_2 * math.pow(x_45_scale, 2.0))))) / math.pi)
	elif b <= -6.4e-43:
		tmp = 180.0 * (math.atan((x_45_scale * (0.005555555555555556 * ((math.pi / x_45_scale) * (angle / (x_45_scale / y_45_scale)))))) / math.pi)
	elif b <= -1.7e-82:
		tmp = t_4
	elif b <= 3e-237:
		tmp = 180.0 * (math.atan((t_3 * (y_45_scale / x_45_scale))) / math.pi)
	elif b <= 1.25e+39:
		tmp = 180.0 * (math.atan((t_0 * (y_45_scale / x_45_scale))) / math.pi)
	else:
		tmp = t_4
	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(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1)
	t_3 = sin(t_0)
	t_4 = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(Float64((cos(t_0) ^ 2.0) / Float64(x_45_scale * x_45_scale)) * Float64(Float64(-y_45_scale) / t_3)))) / pi))
	tmp = 0.0
	if (b <= -2.1e+120)
		tmp = t_4;
	elseif (b <= -4.4e+88)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_2) / x_45_scale)) / pi));
	elseif (b <= -1.75e-6)
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(Float64(y_45_scale * Float64(-(cos(t_1) ^ 2.0))) / Float64(t_2 * (x_45_scale ^ 2.0))))) / pi));
	elseif (b <= -6.4e-43)
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(0.005555555555555556 * Float64(Float64(pi / x_45_scale) * Float64(angle / Float64(x_45_scale / y_45_scale)))))) / pi));
	elseif (b <= -1.7e-82)
		tmp = t_4;
	elseif (b <= 3e-237)
		tmp = Float64(180.0 * Float64(atan(Float64(t_3 * Float64(y_45_scale / x_45_scale))) / pi));
	elseif (b <= 1.25e+39)
		tmp = Float64(180.0 * Float64(atan(Float64(t_0 * Float64(y_45_scale / x_45_scale))) / pi));
	else
		tmp = t_4;
	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 = angle * (pi * 0.005555555555555556);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_1);
	t_3 = sin(t_0);
	t_4 = 180.0 * (atan((x_45_scale * (((cos(t_0) ^ 2.0) / (x_45_scale * x_45_scale)) * (-y_45_scale / t_3)))) / pi);
	tmp = 0.0;
	if (b <= -2.1e+120)
		tmp = t_4;
	elseif (b <= -4.4e+88)
		tmp = 180.0 * (atan(((y_45_scale * t_2) / x_45_scale)) / pi);
	elseif (b <= -1.75e-6)
		tmp = 180.0 * (atan((x_45_scale * ((y_45_scale * -(cos(t_1) ^ 2.0)) / (t_2 * (x_45_scale ^ 2.0))))) / pi);
	elseif (b <= -6.4e-43)
		tmp = 180.0 * (atan((x_45_scale * (0.005555555555555556 * ((pi / x_45_scale) * (angle / (x_45_scale / y_45_scale)))))) / pi);
	elseif (b <= -1.7e-82)
		tmp = t_4;
	elseif (b <= 3e-237)
		tmp = 180.0 * (atan((t_3 * (y_45_scale / x_45_scale))) / pi);
	elseif (b <= 1.25e+39)
		tmp = 180.0 * (atan((t_0 * (y_45_scale / x_45_scale))) / pi);
	else
		tmp = t_4;
	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[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[((-y$45$scale) / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.1e+120], t$95$4, If[LessEqual[b, -4.4e+88], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.75e-6], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(N[(y$45$scale * (-N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / N[(t$95$2 * N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.4e-43], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(0.005555555555555556 * N[(N[(Pi / x$45$scale), $MachinePrecision] * N[(angle / N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.7e-82], t$95$4, If[LessEqual[b, 3e-237], N[(180.0 * N[(N[ArcTan[N[(t$95$3 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e+39], N[(180.0 * N[(N[ArcTan[N[(t$95$0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Bits error versus x-scale

Bits error versus y-scale

Derivation

1. Split input into 6 regimes

2. if b < -2.1e120 or -6.3999999999999997e-43 < b < -1.69999999999999988e-82 or 1.25000000000000004e39 < b

   1. Initial program 58.4

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

   2. Simplified 57.3

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

   3. Taylor expanded in y-scale around inf 56.3

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

   4. Simplified 51.3

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

   5. Taylor expanded in angle around 0 51.3

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

   6. Taylor expanded in b around inf 31.9

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

   7. Simplified 31.7

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

   if -2.1e120 < b < -4.40000000000000017e88

   1. Initial program 50.6

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

   2. Simplified 49.4

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

   3. Taylor expanded in y-scale around inf 44.8

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

   4. Simplified 39.2

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

   5. Taylor expanded in angle around 0 40.7

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

   6. Taylor expanded in b around 0 41.6

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

   7. Simplified 40.8

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

   8. Taylor expanded in x-scale around 0 39.2

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

   if -4.40000000000000017e88 < b < -1.74999999999999997e-6

   1. Initial program 48.5

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

   2. Simplified 46.1

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

   3. Taylor expanded in y-scale around inf 43.4

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

   4. Simplified 38.8

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

   5. Taylor expanded in angle around 0 40.0

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

   6. Taylor expanded in b around inf 38.0

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

   if -1.74999999999999997e-6 < b < -6.3999999999999997e-43

   1. Initial program 49.1

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

   2. Simplified 47.8

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

   3. Taylor expanded in y-scale around inf 43.1

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

   4. Simplified 39.6

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

   5. Taylor expanded in angle around 0 37.8

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

   6. Taylor expanded in b around 0 36.7

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

   7. Simplified 36.0

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

   8. Taylor expanded in angle around 0 37.0

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

   9. Simplified 34.4

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

   if -1.69999999999999988e-82 < b < 3.00000000000000024e-237

   1. Initial program 56.2

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

   2. Simplified 54.0

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

   3. Taylor expanded in y-scale around inf 51.9

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

   4. Simplified 46.3

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

   5. Taylor expanded in angle around 0 46.2

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

   6. Taylor expanded in b around 0 31.1

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

   7. Simplified 30.0

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

   8. Taylor expanded in x-scale around 0 26.3

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

   9. Simplified 24.3

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

   if 3.00000000000000024e-237 < b < 1.25000000000000004e39

   1. Initial program 50.5

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

   2. Simplified 48.5

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

   3. Taylor expanded in y-scale around inf 46.7

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

   4. Simplified 41.2

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

   5. Taylor expanded in angle around 0 42.2

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

   6. Taylor expanded in b around 0 34.8

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

   7. Simplified 33.7

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

   8. Taylor expanded in angle around 0 34.0

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

   9. Simplified 30.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}{\pi} \]
3. Recombined 6 regimes into one program.

4. Final simplification 30.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+120}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale \cdot x-scale} \cdot \frac{-y-scale}{\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{+88}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{y-scale \cdot \left(-{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {x-scale}^{2}}\right)}{\pi}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-43}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(0.005555555555555556 \cdot \left(\frac{\pi}{x-scale} \cdot \frac{angle}{\frac{x-scale}{y-scale}}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale \cdot x-scale} \cdot \frac{-y-scale}{\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-237}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale \cdot x-scale} \cdot \frac{-y-scale}{\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022169 
(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)))