raw-angle from scale-rotated-ellipse

Average Error: 55.1 → 33.7

Time: 1.4min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t_1\\ t_3 := \sin t_1\\ t_4 := \sqrt[3]{t_3}\\ t_5 := \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-{t_2}^{2}}{{x-scale}^{2} \cdot t_3}\right)\right) \cdot \frac{180}{\pi}\\ t_6 := \frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{t_3}{{x-scale}^{2}}\right)\right)\\ t_7 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_8 := {\cos t_7}^{2}\\ t_9 := 2 \cdot \left(b \cdot b - a \cdot a\right)\\ \mathbf{if}\;b \leq -2.3799167193033167 \cdot 10^{+142}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;b \leq -1.5732975136754037 \cdot 10^{+58}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq -6.31696280388902 \cdot 10^{-62}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(\frac{\frac{\mathsf{fma}\left(b \cdot b, t_8, \mathsf{fma}\left(2, {\sin t_7}^{2} \cdot \left(a \cdot a\right), \left(b \cdot b\right) \cdot t_8\right)\right)}{x-scale \cdot x-scale}}{\sin t_0 \cdot \left(\cos t_0 \cdot t_9\right)} \cdot \left(-y-scale\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.920572844790323 \cdot 10^{-138}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq 1.0918947006185081 \cdot 10^{+67}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \left(\frac{-2}{{t_4}^{2}} \cdot \frac{{\left(\mathsf{hypot}\left(t_3 \cdot \frac{a}{x-scale}, t_2 \cdot \frac{b}{x-scale}\right)\right)}^{2}}{t_9 \cdot t_4}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

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 (/ angle 180.0)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1))
        (t_3 (sin t_1))
        (t_4 (cbrt t_3))
        (t_5
         (*
          (atan
           (*
            x-scale
            (* y-scale (/ (- (pow t_2 2.0)) (* (pow x-scale 2.0) t_3)))))
          (/ 180.0 PI)))
        (t_6
         (*
          (/ 180.0 PI)
          (atan (* x-scale (* y-scale (/ t_3 (pow x-scale 2.0)))))))
        (t_7 (* PI (* 0.005555555555555556 angle)))
        (t_8 (pow (cos t_7) 2.0))
        (t_9 (* 2.0 (- (* b b) (* a a)))))
   (if (<= b -2.3799167193033167e+142)
     t_5
     (if (<= b -1.5732975136754037e+58)
       t_6
       (if (<= b -6.31696280388902e-62)
         (*
          (/ 180.0 PI)
          (atan
           (*
            x-scale
            (*
             (/
              (/
               (fma
                (* b b)
                t_8
                (fma 2.0 (* (pow (sin t_7) 2.0) (* a a)) (* (* b b) t_8)))
               (* x-scale x-scale))
              (* (sin t_0) (* (cos t_0) t_9)))
             (- y-scale)))))
         (if (<= b 2.920572844790323e-138)
           t_6
           (if (<= b 1.0918947006185081e+67)
             (*
              (/ 180.0 PI)
              (atan
               (*
                x-scale
                (*
                 y-scale
                 (*
                  (/ -2.0 (pow t_4 2.0))
                  (/
                   (pow
                    (hypot (* t_3 (/ a x-scale)) (* t_2 (/ b x-scale)))
                    2.0)
                   (* t_9 t_4)))))))
             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) * (angle / 180.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = sin(t_1);
	double t_4 = cbrt(t_3);
	double t_5 = atan((x_45_scale * (y_45_scale * (-pow(t_2, 2.0) / (pow(x_45_scale, 2.0) * t_3))))) * (180.0 / ((double) M_PI));
	double t_6 = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * (t_3 / pow(x_45_scale, 2.0)))));
	double t_7 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_8 = pow(cos(t_7), 2.0);
	double t_9 = 2.0 * ((b * b) - (a * a));
	double tmp;
	if (b <= -2.3799167193033167e+142) {
		tmp = t_5;
	} else if (b <= -1.5732975136754037e+58) {
		tmp = t_6;
	} else if (b <= -6.31696280388902e-62) {
		tmp = (180.0 / ((double) M_PI)) * atan((x_45_scale * (((fma((b * b), t_8, fma(2.0, (pow(sin(t_7), 2.0) * (a * a)), ((b * b) * t_8))) / (x_45_scale * x_45_scale)) / (sin(t_0) * (cos(t_0) * t_9))) * -y_45_scale)));
	} else if (b <= 2.920572844790323e-138) {
		tmp = t_6;
	} else if (b <= 1.0918947006185081e+67) {
		tmp = (180.0 / ((double) M_PI)) * atan((x_45_scale * (y_45_scale * ((-2.0 / pow(t_4, 2.0)) * (pow(hypot((t_3 * (a / x_45_scale)), (t_2 * (b / x_45_scale))), 2.0) / (t_9 * t_4))))));
	} 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(angle / 180.0))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = sin(t_1)
	t_4 = cbrt(t_3)
	t_5 = Float64(atan(Float64(x_45_scale * Float64(y_45_scale * Float64(Float64(-(t_2 ^ 2.0)) / Float64((x_45_scale ^ 2.0) * t_3))))) * Float64(180.0 / pi))
	t_6 = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * Float64(t_3 / (x_45_scale ^ 2.0))))))
	t_7 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_8 = cos(t_7) ^ 2.0
	t_9 = Float64(2.0 * Float64(Float64(b * b) - Float64(a * a)))
	tmp = 0.0
	if (b <= -2.3799167193033167e+142)
		tmp = t_5;
	elseif (b <= -1.5732975136754037e+58)
		tmp = t_6;
	elseif (b <= -6.31696280388902e-62)
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(Float64(Float64(fma(Float64(b * b), t_8, fma(2.0, Float64((sin(t_7) ^ 2.0) * Float64(a * a)), Float64(Float64(b * b) * t_8))) / Float64(x_45_scale * x_45_scale)) / Float64(sin(t_0) * Float64(cos(t_0) * t_9))) * Float64(-y_45_scale)))));
	elseif (b <= 2.920572844790323e-138)
		tmp = t_6;
	elseif (b <= 1.0918947006185081e+67)
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(x_45_scale * Float64(y_45_scale * Float64(Float64(-2.0 / (t_4 ^ 2.0)) * Float64((hypot(Float64(t_3 * Float64(a / x_45_scale)), Float64(t_2 * Float64(b / x_45_scale))) ^ 2.0) / Float64(t_9 * t_4)))))));
	else
		tmp = t_5;
	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[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 1/3], $MachinePrecision]}, Block[{t$95$5 = N[(N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[((-N[Power[t$95$2, 2.0], $MachinePrecision]) / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * t$95$3), $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$3 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Power[N[Cos[t$95$7], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$9 = N[(2.0 * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.3799167193033167e+142], t$95$5, If[LessEqual[b, -1.5732975136754037e+58], t$95$6, If[LessEqual[b, -6.31696280388902e-62], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(N[(N[(N[(N[(b * b), $MachinePrecision] * t$95$8 + N[(2.0 * N[(N[Power[N[Sin[t$95$7], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-y$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.920572844790323e-138], t$95$6, If[LessEqual[b, 1.0918947006185081e+67], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(x$45$scale * N[(y$45$scale * N[(N[(-2.0 / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sqrt[N[(t$95$3 * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(t$95$2 * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$9 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.3799167193033167e142 or 1.09189470061850811e67 < b

   1. Initial program 60.6

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

   2. Simplified 59.7

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

   3. Taylor expanded in y-scale around inf 58.8

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

   4. Simplified 53.6

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

   5. Applied egg-rr 53.6

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

   6. Taylor expanded in angle around 0 53.4

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

   7. Taylor expanded in b around inf 30.3

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

   if -2.3799167193033167e142 < b < -1.5732975136754037e58 or -6.31696280388901975e-62 < b < 2.9205728447903231e-138

   1. Initial program 55.0

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

   2. Simplified 52.8

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

   3. Taylor expanded in y-scale around inf 49.4

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

   4. Simplified 41.3

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

   5. Applied egg-rr 41.1

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

   6. Taylor expanded in angle around 0 41.9

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

   7. Taylor expanded in b around 0 33.7

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

   if -1.5732975136754037e58 < b < -6.31696280388901975e-62

   1. Initial program 47.5

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

   2. Simplified 45.6

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

   3. Taylor expanded in x-scale around 0 38.1

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

   4. Simplified 38.1

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

   if 2.9205728447903231e-138 < b < 1.09189470061850811e67

   1. Initial program 48.7

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

   2. Simplified 46.7

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

   3. Taylor expanded in y-scale around inf 43.8

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

   4. Simplified 38.7

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

   5. Applied egg-rr 38.8

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

   6. Taylor expanded in angle around 0 39.7

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

   7. Applied egg-rr 37.7

      \[\leadsto \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \color{blue}{\left(\frac{-2}{{\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2}} \cdot \frac{{\left(\mathsf{hypot}\left(\frac{a}{x-scale} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \frac{b}{x-scale} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}{\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}\right)\right) \cdot \frac{180}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3799167193033167 \cdot 10^{+142}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-{\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) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;b \leq -1.5732975136754037 \cdot 10^{+58}:\\ \;\;\;\;\frac{180}{\pi} \cdot \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}}\right)\right)\\ \mathbf{elif}\;b \leq -6.31696280388902 \cdot 10^{-62}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(\frac{\frac{\mathsf{fma}\left(b \cdot b, {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}, \mathsf{fma}\left(2, {\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right), \left(b \cdot b\right) \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}\right)\right)}{x-scale \cdot x-scale}}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)} \cdot \left(-y-scale\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.920572844790323 \cdot 10^{-138}:\\ \;\;\;\;\frac{180}{\pi} \cdot \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}}\right)\right)\\ \mathbf{elif}\;b \leq 1.0918947006185081 \cdot 10^{+67}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \left(\frac{-2}{{\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2}} \cdot \frac{{\left(\mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{a}{x-scale}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{b}{x-scale}\right)\right)}^{2}}{\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(x-scale \cdot \left(y-scale \cdot \frac{-{\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) \cdot \frac{180}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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)))