raw-angle from scale-rotated-ellipse

Average Error: 55.3 → 30.4

Time: 1.7min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{y-scale \cdot \left(-{\cos t_1}^{2}\right)}{{x-scale}^{2} \cdot \sin t_1}\right)}{\pi}\\ \mathbf{if}\;b \leq -2.6352289852693054 \cdot 10^{+117}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{-y-scale}{x-scale \cdot x-scale} \cdot \frac{{\cos t_0}^{2}}{\sin t_0}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -3.1874558472003755 \cdot 10^{+23}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, {\pi}^{3} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot {angle}^{3}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -4.138763665321299 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 8.578813038149054 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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
         (*
          180.0
          (/
           (atan
            (*
             x-scale
             (/
              (* y-scale (- (pow (cos t_1) 2.0)))
              (* (pow x-scale 2.0) (sin t_1)))))
           PI))))
   (if (<= b -2.6352289852693054e+117)
     (*
      180.0
      (/
       (atan
        (*
         x-scale
         (*
          (/ (- y-scale) (* x-scale x-scale))
          (/ (pow (cos t_0) 2.0) (sin t_0)))))
       PI))
     (if (<= b -3.1874558472003755e+23)
       (*
        180.0
        (/
         (atan
          (*
           (/ y-scale x-scale)
           (fma
            0.005555555555555556
            (* angle PI)
            (* (pow PI 3.0) (* -2.8577960676726107e-8 (pow angle 3.0))))))
         PI))
       (if (<= b -4.138763665321299e-35)
         t_2
         (if (<= b 8.578813038149054e-52)
           (*
            180.0
            (/
             (atan
              (*
               (/ y-scale x-scale)
               (sin (* PI (* angle 0.005555555555555556)))))
             PI))
           t_2))))))

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 = 180.0 * (atan((x_45_scale * ((y_45_scale * -pow(cos(t_1), 2.0)) / (pow(x_45_scale, 2.0) * sin(t_1))))) / ((double) M_PI));
	double tmp;
	if (b <= -2.6352289852693054e+117) {
		tmp = 180.0 * (atan((x_45_scale * ((-y_45_scale / (x_45_scale * x_45_scale)) * (pow(cos(t_0), 2.0) / sin(t_0))))) / ((double) M_PI));
	} else if (b <= -3.1874558472003755e+23) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * fma(0.005555555555555556, (angle * ((double) M_PI)), (pow(((double) M_PI), 3.0) * (-2.8577960676726107e-8 * pow(angle, 3.0)))))) / ((double) M_PI));
	} else if (b <= -4.138763665321299e-35) {
		tmp = t_2;
	} else if (b <= 8.578813038149054e-52) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((((double) M_PI) * (angle * 0.005555555555555556))))) / ((double) M_PI));
	} else {
		tmp = t_2;
	}
	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 = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(Float64(y_45_scale * Float64(-(cos(t_1) ^ 2.0))) / Float64((x_45_scale ^ 2.0) * sin(t_1))))) / pi))
	tmp = 0.0
	if (b <= -2.6352289852693054e+117)
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(Float64(Float64(-y_45_scale) / Float64(x_45_scale * x_45_scale)) * Float64((cos(t_0) ^ 2.0) / sin(t_0))))) / pi));
	elseif (b <= -3.1874558472003755e+23)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * fma(0.005555555555555556, Float64(angle * pi), Float64((pi ^ 3.0) * Float64(-2.8577960676726107e-8 * (angle ^ 3.0)))))) / pi));
	elseif (b <= -4.138763665321299e-35)
		tmp = t_2;
	elseif (b <= 8.578813038149054e-52)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(Float64(pi * Float64(angle * 0.005555555555555556))))) / pi));
	else
		tmp = t_2;
	end
	return 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[(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[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.6352289852693054e+117], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(N[((-y$45$scale) / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.1874558472003755e+23], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(-2.8577960676726107e-8 * N[Power[angle, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.138763665321299e-35], t$95$2, If[LessEqual[b, 8.578813038149054e-52], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.63522898526930538e117

   1. Initial program 61.6

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

   2. Simplified 60.9

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

   3. Taylor expanded in y-scale around inf 60.2

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

   4. Simplified 55.5

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

   5. Taylor expanded in angle around 0 55.6

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

   6. Taylor expanded in b around inf 30.9

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

   7. Simplified 31.5

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

   if -2.63522898526930538e117 < b < -3.1874558472003755e23

   1. Initial program 48.9

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

   2. Simplified 45.7

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

   3. Taylor expanded in y-scale around inf 44.3

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

   4. Simplified 36.6

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

   5. Taylor expanded in angle around 0 37.2

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

   6. Taylor expanded in b around 0 40.4

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

   7. Simplified 39.2

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

   8. Taylor expanded in angle around 0 48.7

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-2.8577960676726107 \cdot 10^{-8} \cdot \frac{y-scale \cdot \left({angle}^{3} \cdot {\pi}^{3}\right)}{x-scale} + 0.005555555555555556 \cdot \frac{y-scale \cdot \left(angle \cdot \pi\right)}{x-scale}\right)}}{\pi} \]

   9. Simplified 40.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, {\pi}^{3} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot {angle}^{3}\right)\right)\right)}}{\pi} \]

   if -3.1874558472003755e23 < b < -4.13876366532129883e-35 or 8.5788130381490539e-52 < b

   1. Initial program 54.2

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

   2. Simplified 52.6

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

   3. Taylor expanded in y-scale around inf 50.8

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

   4. Simplified 46.0

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

   5. Taylor expanded in angle around 0 46.1

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

   6. Taylor expanded in b around inf 33.0

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

   if -4.13876366532129883e-35 < b < 8.5788130381490539e-52

   1. Initial program 54.8

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

   2. Simplified 53.2

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

   3. Taylor expanded in y-scale around inf 50.7

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

   4. Simplified 45.0

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

   5. Taylor expanded in angle around 0 45.0

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

   6. Taylor expanded in b around 0 31.9

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

   7. Simplified 31.1

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

   8. Taylor expanded in x-scale around 0 27.8

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

   9. Simplified 26.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{y-scale}{x-scale}\right)}}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6352289852693054 \cdot 10^{+117}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(\frac{-y-scale}{x-scale \cdot x-scale} \cdot \frac{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}{\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -3.1874558472003755 \cdot 10^{+23}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, {\pi}^{3} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot {angle}^{3}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -4.138763665321299 \cdot 10^{-35}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{y-scale \cdot \left(-{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 8.578813038149054 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{y-scale \cdot \left(-{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022204 
(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)))