raw-angle from scale-rotated-ellipse

Average Error: 55.0 → 37.7

Time: 1.6min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t_1\\ t_3 := \cos t_0\\ t_4 := \cos t_1\\ t_5 := \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{t_2}{{x-scale}^{2} \cdot t_4}\right)\right) \cdot \frac{180}{\pi}\\ t_6 := \frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-t_4}{t_2 \cdot {x-scale}^{2}}\right)\right)\\ t_7 := \sin t_0\\ t_8 := \frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-t_3}{t_7 \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\\ t_9 := \frac{t_7}{x-scale \cdot \left(x-scale \cdot t_3\right)}\\ t_10 := \left(x-scale \cdot a\right) \cdot \left(x-scale \cdot a\right)\\ \mathbf{if}\;x-scale \leq -2.3273358225878017 \cdot 10^{+33}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x-scale \leq -1.7544344154926862 \cdot 10^{-68}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;x-scale \leq -4.534307734857159 \cdot 10^{-308}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x-scale \leq 6.895664114487812 \cdot 10^{-139}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x-scale \leq 2.4232374911042355 \cdot 10^{-95}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x-scale \leq 1.1138736257067423 \cdot 10^{-41}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;x-scale \leq 2.0880495689654187 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot t_9\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.8347736864239356 \cdot 10^{+50}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x-scale \leq 9.063302108862638 \cdot 10^{+81}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \left(t_9 + \left(b \cdot b\right) \cdot \left(\frac{\frac{t_3}{t_10}}{t_7} + \frac{t_7}{t_3 \cdot t_10}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_1))
        (t_3 (cos t_0))
        (t_4 (cos t_1))
        (t_5
         (*
          (atan (* x-scale (* y-scale (/ t_2 (* (pow x-scale 2.0) t_4)))))
          (/ 180.0 PI)))
        (t_6
         (*
          (/ 180.0 PI)
          (atan
           (* x-scale (* y-scale (/ (- t_4) (* t_2 (pow x-scale 2.0))))))))
        (t_7 (sin t_0))
        (t_8
         (*
          (/ 180.0 PI)
          (atan
           (* x-scale (* y-scale (/ (- t_3) (* t_7 (* x-scale x-scale))))))))
        (t_9 (/ t_7 (* x-scale (* x-scale t_3))))
        (t_10 (* (* x-scale a) (* x-scale a))))
   (if (<= x-scale -2.3273358225878017e+33)
     t_5
     (if (<= x-scale -1.7544344154926862e-68)
       t_8
       (if (<= x-scale -4.534307734857159e-308)
         t_5
         (if (<= x-scale 6.895664114487812e-139)
           t_6
           (if (<= x-scale 2.4232374911042355e-95)
             t_5
             (if (<= x-scale 1.1138736257067423e-41)
               t_8
               (if (<= x-scale 2.0880495689654187e-9)
                 (* (/ 180.0 PI) (atan (* x-scale (* y-scale t_9))))
                 (if (<= x-scale 1.8347736864239356e+50)
                   t_6
                   (if (<= x-scale 9.063302108862638e+81)
                     (*
                      (/ 180.0 PI)
                      (atan
                       (*
                        x-scale
                        (*
                         y-scale
                         (+
                          t_9
                          (*
                           (* b b)
                           (+ (/ (/ t_3 t_10) t_7) (/ t_7 (* t_3 t_10)))))))))
                     t_5)))))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_1);
	double t_3 = cos(t_0);
	double t_4 = cos(t_1);
	double t_5 = atan((x_45_scale * (y_45_scale * (t_2 / (pow(x_45_scale, 2.0) * t_4))))) * (180.0 / ((double) M_PI));
	double t_6 = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * (-t_4 / (t_2 * pow(x_45_scale, 2.0))))));
	double t_7 = sin(t_0);
	double t_8 = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * (-t_3 / (t_7 * (x_45_scale * x_45_scale))))));
	double t_9 = t_7 / (x_45_scale * (x_45_scale * t_3));
	double t_10 = (x_45_scale * a) * (x_45_scale * a);
	double tmp;
	if (x_45_scale <= -2.3273358225878017e+33) {
		tmp = t_5;
	} else if (x_45_scale <= -1.7544344154926862e-68) {
		tmp = t_8;
	} else if (x_45_scale <= -4.534307734857159e-308) {
		tmp = t_5;
	} else if (x_45_scale <= 6.895664114487812e-139) {
		tmp = t_6;
	} else if (x_45_scale <= 2.4232374911042355e-95) {
		tmp = t_5;
	} else if (x_45_scale <= 1.1138736257067423e-41) {
		tmp = t_8;
	} else if (x_45_scale <= 2.0880495689654187e-9) {
		tmp = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * t_9)));
	} else if (x_45_scale <= 1.8347736864239356e+50) {
		tmp = t_6;
	} else if (x_45_scale <= 9.063302108862638e+81) {
		tmp = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * (t_9 + ((b * b) * (((t_3 / t_10) / t_7) + (t_7 / (t_3 * t_10))))))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.sin(t_1);
	double t_3 = Math.cos(t_0);
	double t_4 = Math.cos(t_1);
	double t_5 = Math.atan((x_45_scale * (y_45_scale * (t_2 / (Math.pow(x_45_scale, 2.0) * t_4))))) * (180.0 / Math.PI);
	double t_6 = (180.0 / Math.PI) * Math.atan((x_45_scale * (y_45_scale * (-t_4 / (t_2 * Math.pow(x_45_scale, 2.0))))));
	double t_7 = Math.sin(t_0);
	double t_8 = (180.0 / Math.PI) * Math.atan((x_45_scale * (y_45_scale * (-t_3 / (t_7 * (x_45_scale * x_45_scale))))));
	double t_9 = t_7 / (x_45_scale * (x_45_scale * t_3));
	double t_10 = (x_45_scale * a) * (x_45_scale * a);
	double tmp;
	if (x_45_scale <= -2.3273358225878017e+33) {
		tmp = t_5;
	} else if (x_45_scale <= -1.7544344154926862e-68) {
		tmp = t_8;
	} else if (x_45_scale <= -4.534307734857159e-308) {
		tmp = t_5;
	} else if (x_45_scale <= 6.895664114487812e-139) {
		tmp = t_6;
	} else if (x_45_scale <= 2.4232374911042355e-95) {
		tmp = t_5;
	} else if (x_45_scale <= 1.1138736257067423e-41) {
		tmp = t_8;
	} else if (x_45_scale <= 2.0880495689654187e-9) {
		tmp = (180.0 / Math.PI) * Math.atan((x_45_scale * (y_45_scale * t_9)));
	} else if (x_45_scale <= 1.8347736864239356e+50) {
		tmp = t_6;
	} else if (x_45_scale <= 9.063302108862638e+81) {
		tmp = (180.0 / Math.PI) * Math.atan((x_45_scale * (y_45_scale * (t_9 + ((b * b) * (((t_3 / t_10) / t_7) + (t_7 / (t_3 * t_10))))))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.sin(t_1)
	t_3 = math.cos(t_0)
	t_4 = math.cos(t_1)
	t_5 = math.atan((x_45_scale * (y_45_scale * (t_2 / (math.pow(x_45_scale, 2.0) * t_4))))) * (180.0 / math.pi)
	t_6 = (180.0 / math.pi) * math.atan((x_45_scale * (y_45_scale * (-t_4 / (t_2 * math.pow(x_45_scale, 2.0))))))
	t_7 = math.sin(t_0)
	t_8 = (180.0 / math.pi) * math.atan((x_45_scale * (y_45_scale * (-t_3 / (t_7 * (x_45_scale * x_45_scale))))))
	t_9 = t_7 / (x_45_scale * (x_45_scale * t_3))
	t_10 = (x_45_scale * a) * (x_45_scale * a)
	tmp = 0
	if x_45_scale <= -2.3273358225878017e+33:
		tmp = t_5
	elif x_45_scale <= -1.7544344154926862e-68:
		tmp = t_8
	elif x_45_scale <= -4.534307734857159e-308:
		tmp = t_5
	elif x_45_scale <= 6.895664114487812e-139:
		tmp = t_6
	elif x_45_scale <= 2.4232374911042355e-95:
		tmp = t_5
	elif x_45_scale <= 1.1138736257067423e-41:
		tmp = t_8
	elif x_45_scale <= 2.0880495689654187e-9:
		tmp = (180.0 / math.pi) * math.atan((x_45_scale * (y_45_scale * t_9)))
	elif x_45_scale <= 1.8347736864239356e+50:
		tmp = t_6
	elif x_45_scale <= 9.063302108862638e+81:
		tmp = (180.0 / math.pi) * math.atan((x_45_scale * (y_45_scale * (t_9 + ((b * b) * (((t_3 / t_10) / t_7) + (t_7 / (t_3 * t_10))))))))
	else:
		tmp = t_5
	return tmp

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1)
	t_3 = cos(t_0)
	t_4 = cos(t_1)
	t_5 = Float64(atan(Float64(x_45_scale * Float64(y_45_scale * Float64(t_2 / Float64((x_45_scale ^ 2.0) * t_4))))) * Float64(180.0 / pi))
	t_6 = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * Float64(Float64(-t_4) / Float64(t_2 * (x_45_scale ^ 2.0)))))))
	t_7 = sin(t_0)
	t_8 = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * Float64(Float64(-t_3) / Float64(t_7 * Float64(x_45_scale * x_45_scale)))))))
	t_9 = Float64(t_7 / Float64(x_45_scale * Float64(x_45_scale * t_3)))
	t_10 = Float64(Float64(x_45_scale * a) * Float64(x_45_scale * a))
	tmp = 0.0
	if (x_45_scale <= -2.3273358225878017e+33)
		tmp = t_5;
	elseif (x_45_scale <= -1.7544344154926862e-68)
		tmp = t_8;
	elseif (x_45_scale <= -4.534307734857159e-308)
		tmp = t_5;
	elseif (x_45_scale <= 6.895664114487812e-139)
		tmp = t_6;
	elseif (x_45_scale <= 2.4232374911042355e-95)
		tmp = t_5;
	elseif (x_45_scale <= 1.1138736257067423e-41)
		tmp = t_8;
	elseif (x_45_scale <= 2.0880495689654187e-9)
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * t_9))));
	elseif (x_45_scale <= 1.8347736864239356e+50)
		tmp = t_6;
	elseif (x_45_scale <= 9.063302108862638e+81)
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * Float64(t_9 + Float64(Float64(b * b) * Float64(Float64(Float64(t_3 / t_10) / t_7) + Float64(t_7 / Float64(t_3 * t_10)))))))));
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_1);
	t_3 = cos(t_0);
	t_4 = cos(t_1);
	t_5 = atan((x_45_scale * (y_45_scale * (t_2 / ((x_45_scale ^ 2.0) * t_4))))) * (180.0 / pi);
	t_6 = (180.0 / pi) * atan((x_45_scale * (y_45_scale * (-t_4 / (t_2 * (x_45_scale ^ 2.0))))));
	t_7 = sin(t_0);
	t_8 = (180.0 / pi) * atan((x_45_scale * (y_45_scale * (-t_3 / (t_7 * (x_45_scale * x_45_scale))))));
	t_9 = t_7 / (x_45_scale * (x_45_scale * t_3));
	t_10 = (x_45_scale * a) * (x_45_scale * a);
	tmp = 0.0;
	if (x_45_scale <= -2.3273358225878017e+33)
		tmp = t_5;
	elseif (x_45_scale <= -1.7544344154926862e-68)
		tmp = t_8;
	elseif (x_45_scale <= -4.534307734857159e-308)
		tmp = t_5;
	elseif (x_45_scale <= 6.895664114487812e-139)
		tmp = t_6;
	elseif (x_45_scale <= 2.4232374911042355e-95)
		tmp = t_5;
	elseif (x_45_scale <= 1.1138736257067423e-41)
		tmp = t_8;
	elseif (x_45_scale <= 2.0880495689654187e-9)
		tmp = (180.0 / pi) * atan((x_45_scale * (y_45_scale * t_9)));
	elseif (x_45_scale <= 1.8347736864239356e+50)
		tmp = t_6;
	elseif (x_45_scale <= 9.063302108862638e+81)
		tmp = (180.0 / pi) * atan((x_45_scale * (y_45_scale * (t_9 + ((b * b) * (((t_3 / t_10) / t_7) + (t_7 / (t_3 * t_10))))))));
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[(t$95$2 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[((-t$95$4) / N[(t$95$2 * N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$8 = N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[((-t$95$3) / N[(t$95$7 * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$7 / N[(x$45$scale * N[(x$45$scale * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(x$45$scale * a), $MachinePrecision] * N[(x$45$scale * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2.3273358225878017e+33], t$95$5, If[LessEqual[x$45$scale, -1.7544344154926862e-68], t$95$8, If[LessEqual[x$45$scale, -4.534307734857159e-308], t$95$5, If[LessEqual[x$45$scale, 6.895664114487812e-139], t$95$6, If[LessEqual[x$45$scale, 2.4232374911042355e-95], t$95$5, If[LessEqual[x$45$scale, 1.1138736257067423e-41], t$95$8, If[LessEqual[x$45$scale, 2.0880495689654187e-9], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(y$45$scale * t$95$9), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 1.8347736864239356e+50], t$95$6, If[LessEqual[x$45$scale, 9.063302108862638e+81], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[(t$95$9 + N[(N[(b * b), $MachinePrecision] * N[(N[(N[(t$95$3 / t$95$10), $MachinePrecision] / t$95$7), $MachinePrecision] + N[(t$95$7 / N[(t$95$3 * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x-scale < -2.3273358225878017e33 or -1.7544344154926862e-68 < x-scale < -4.53430773485715935e-308 or 6.8956641144878118e-139 < x-scale < 2.423237491104236e-95 or 9.0633021088626383e81 < x-scale

   1. Initial program 56.7

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

   2. Simplified 55.0

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

   3. Taylor expanded in y-scale around inf 51.0

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

   4. Simplified 43.3

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

   5. Taylor expanded in angle around inf 43.2

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

   6. Applied egg-rr 43.2

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} + \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)}}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}\right)\right) \cdot \frac{180}{\pi} \]

   7. Taylor expanded in b around 0 37.6

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

   if -2.3273358225878017e33 < x-scale < -1.7544344154926862e-68 or 2.423237491104236e-95 < x-scale < 1.1138736257067423e-41

   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 50.1

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

   3. Taylor expanded in y-scale around inf 46.1

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

   4. Simplified 46.1

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

   5. Taylor expanded in angle around inf 46.3

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

   6. Applied egg-rr 46.4

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} + \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)}}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}\right)\right) \cdot \frac{180}{\pi} \]

   7. Taylor expanded in b around inf 39.6

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

   8. Simplified 38.9

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

   if -4.53430773485715935e-308 < x-scale < 6.8956641144878118e-139 or 2.0880495689654187e-9 < x-scale < 1.8347736864239356e50

   1. Initial program 52.6

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

   2. Simplified 49.2

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

   3. Taylor expanded in y-scale around inf 53.2

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

   4. Simplified 44.5

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

   5. Taylor expanded in angle around inf 44.8

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

   6. Applied egg-rr 44.8

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} + \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)}}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}\right)\right) \cdot \frac{180}{\pi} \]

   7. Taylor expanded in b around inf 36.1

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

   if 1.1138736257067423e-41 < x-scale < 2.0880495689654187e-9

   1. Initial program 52.1

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

   2. Simplified 51.8

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

   3. Taylor expanded in y-scale around inf 47.2

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

   4. Simplified 46.9

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

   5. Taylor expanded in angle around inf 47.1

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

   6. Applied egg-rr 47.2

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} + \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)}}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}\right)\right) \cdot \frac{180}{\pi} \]

   7. Taylor expanded in b around 0 37.7

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

   8. Simplified 37.5

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

   if 1.8347736864239356e50 < x-scale < 9.0633021088626383e81

   1. Initial program 53.6

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

   2. Simplified 53.2

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

   3. Taylor expanded in y-scale around inf 46.1

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

   4. Simplified 44.4

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

   5. Taylor expanded in angle around inf 45.0

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

   6. Applied egg-rr 45.0

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} + \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)}}^{2}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}\right)\right) \cdot \frac{180}{\pi} \]

   7. Taylor expanded in b around 0 44.8

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

   8. Simplified 43.4

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

4. Final simplification 37.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -2.3273358225878017 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;x-scale \leq -1.7544344154926862 \cdot 10^{-68}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\\ \mathbf{elif}\;x-scale \leq -4.534307734857159 \cdot 10^{-308}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;x-scale \leq 6.895664114487812 \cdot 10^{-139}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-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) \cdot {x-scale}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.4232374911042355 \cdot 10^{-95}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;x-scale \leq 1.1138736257067423 \cdot 10^{-41}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.0880495689654187 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale \cdot \left(x-scale \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.8347736864239356 \cdot 10^{+50}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-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) \cdot {x-scale}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq 9.063302108862638 \cdot 10^{+81}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \left(\frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale \cdot \left(x-scale \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + \left(b \cdot b\right) \cdot \left(\frac{\frac{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\left(x-scale \cdot a\right) \cdot \left(x-scale \cdot a\right)}}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} + \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\left(x-scale \cdot a\right) \cdot \left(x-scale \cdot a\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \frac{180}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))