raw-angle from scale-rotated-ellipse

Average Error: 55.2 → 28.8

Time: 1.9min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \cos t_0\\ t_2 := \sin t_0\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale}{\frac{t_2 \cdot \left(x-scale \cdot x-scale\right)}{t_1}}\right)\right)\right)}{\pi}\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_5 := \sin t_4\\ t_6 := \cos t_4\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+128}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t_5 \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-286}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t_5 \cdot \frac{\frac{y-scale}{\cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)}}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-12}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{t_2}{x-scale \cdot \frac{t_1}{y-scale}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+78}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t_5 \cdot \frac{\frac{y-scale}{t_6}}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale \cdot t_6}{t_5 \cdot {x-scale}^{2}}\right)\right)\right)}{\pi}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (*
          180.0
          (/
           (atan
            (*
             x-scale
             (* -0.5 (* 2.0 (/ y-scale (/ (* t_2 (* x-scale x-scale)) t_1))))))
           PI)))
        (t_4 (* 0.005555555555555556 (* angle PI)))
        (t_5 (sin t_4))
        (t_6 (cos t_4)))
   (if (<= b -2.8e+128)
     t_3
     (if (<= b -5.2e+36)
       (* 180.0 (/ (atan (* t_5 (/ y-scale x-scale))) PI))
       (if (<= b -1.2e+22)
         t_3
         (if (<= b -9e-286)
           (*
            180.0
            (/
             (atan (* t_5 (/ (/ y-scale (cos (pow (cbrt t_0) 3.0))) x-scale)))
             PI))
           (if (<= b 6.7e-12)
             (* 180.0 (/ (atan (/ t_2 (* x-scale (/ t_1 y-scale)))) PI))
             (if (<= b 6.6e+46)
               t_3
               (if (<= b 2.5e+78)
                 (* 180.0 (/ (atan (* t_5 (/ (/ y-scale t_6) x-scale))) PI))
                 (*
                  180.0
                  (/
                   (atan
                    (*
                     x-scale
                     (*
                      -0.5
                      (* 2.0 (/ (* y-scale t_6) (* t_5 (pow x-scale 2.0)))))))
                   PI)))))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = 180.0 * (atan((x_45_scale * (-0.5 * (2.0 * (y_45_scale / ((t_2 * (x_45_scale * x_45_scale)) / t_1)))))) / ((double) M_PI));
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_5 = sin(t_4);
	double t_6 = cos(t_4);
	double tmp;
	if (b <= -2.8e+128) {
		tmp = t_3;
	} else if (b <= -5.2e+36) {
		tmp = 180.0 * (atan((t_5 * (y_45_scale / x_45_scale))) / ((double) M_PI));
	} else if (b <= -1.2e+22) {
		tmp = t_3;
	} else if (b <= -9e-286) {
		tmp = 180.0 * (atan((t_5 * ((y_45_scale / cos(pow(cbrt(t_0), 3.0))) / x_45_scale))) / ((double) M_PI));
	} else if (b <= 6.7e-12) {
		tmp = 180.0 * (atan((t_2 / (x_45_scale * (t_1 / y_45_scale)))) / ((double) M_PI));
	} else if (b <= 6.6e+46) {
		tmp = t_3;
	} else if (b <= 2.5e+78) {
		tmp = 180.0 * (atan((t_5 * ((y_45_scale / t_6) / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((x_45_scale * (-0.5 * (2.0 * ((y_45_scale * t_6) / (t_5 * pow(x_45_scale, 2.0))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * Math.PI;
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double t_3 = 180.0 * (Math.atan((x_45_scale * (-0.5 * (2.0 * (y_45_scale / ((t_2 * (x_45_scale * x_45_scale)) / t_1)))))) / Math.PI);
	double t_4 = 0.005555555555555556 * (angle * Math.PI);
	double t_5 = Math.sin(t_4);
	double t_6 = Math.cos(t_4);
	double tmp;
	if (b <= -2.8e+128) {
		tmp = t_3;
	} else if (b <= -5.2e+36) {
		tmp = 180.0 * (Math.atan((t_5 * (y_45_scale / x_45_scale))) / Math.PI);
	} else if (b <= -1.2e+22) {
		tmp = t_3;
	} else if (b <= -9e-286) {
		tmp = 180.0 * (Math.atan((t_5 * ((y_45_scale / Math.cos(Math.pow(Math.cbrt(t_0), 3.0))) / x_45_scale))) / Math.PI);
	} else if (b <= 6.7e-12) {
		tmp = 180.0 * (Math.atan((t_2 / (x_45_scale * (t_1 / y_45_scale)))) / Math.PI);
	} else if (b <= 6.6e+46) {
		tmp = t_3;
	} else if (b <= 2.5e+78) {
		tmp = 180.0 * (Math.atan((t_5 * ((y_45_scale / t_6) / x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((x_45_scale * (-0.5 * (2.0 * ((y_45_scale * t_6) / (t_5 * Math.pow(x_45_scale, 2.0))))))) / Math.PI);
	}
	return tmp;
}

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(-0.5 * Float64(2.0 * Float64(y_45_scale / Float64(Float64(t_2 * Float64(x_45_scale * x_45_scale)) / t_1)))))) / pi))
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_5 = sin(t_4)
	t_6 = cos(t_4)
	tmp = 0.0
	if (b <= -2.8e+128)
		tmp = t_3;
	elseif (b <= -5.2e+36)
		tmp = Float64(180.0 * Float64(atan(Float64(t_5 * Float64(y_45_scale / x_45_scale))) / pi));
	elseif (b <= -1.2e+22)
		tmp = t_3;
	elseif (b <= -9e-286)
		tmp = Float64(180.0 * Float64(atan(Float64(t_5 * Float64(Float64(y_45_scale / cos((cbrt(t_0) ^ 3.0))) / x_45_scale))) / pi));
	elseif (b <= 6.7e-12)
		tmp = Float64(180.0 * Float64(atan(Float64(t_2 / Float64(x_45_scale * Float64(t_1 / y_45_scale)))) / pi));
	elseif (b <= 6.6e+46)
		tmp = t_3;
	elseif (b <= 2.5e+78)
		tmp = Float64(180.0 * Float64(atan(Float64(t_5 * Float64(Float64(y_45_scale / t_6) / x_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(-0.5 * Float64(2.0 * Float64(Float64(y_45_scale * t_6) / Float64(t_5 * (x_45_scale ^ 2.0))))))) / pi));
	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[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(-0.5 * N[(2.0 * N[(y$45$scale / N[(N[(t$95$2 * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Cos[t$95$4], $MachinePrecision]}, If[LessEqual[b, -2.8e+128], t$95$3, If[LessEqual[b, -5.2e+36], N[(180.0 * N[(N[ArcTan[N[(t$95$5 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.2e+22], t$95$3, If[LessEqual[b, -9e-286], N[(180.0 * N[(N[ArcTan[N[(t$95$5 * N[(N[(y$45$scale / N[Cos[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.7e-12], N[(180.0 * N[(N[ArcTan[N[(t$95$2 / N[(x$45$scale * N[(t$95$1 / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e+46], t$95$3, If[LessEqual[b, 2.5e+78], N[(180.0 * N[(N[ArcTan[N[(t$95$5 * N[(N[(y$45$scale / t$95$6), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(-0.5 * N[(2.0 * N[(N[(y$45$scale * t$95$6), $MachinePrecision] / N[(t$95$5 * N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Bits error versus x-scale

Bits error versus y-scale

Derivation

1. Split input into 6 regimes

2. if b < -2.79999999999999983e128 or -5.2000000000000003e36 < b < -1.2e22 or 6.7000000000000001e-12 < b < 6.5999999999999996e46

   1. Initial program 57.1

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

   2. Simplified 55.9

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

   3. Taylor expanded in x-scale around 0 54.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around 0 30.8

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

   5. Simplified 30.2

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

   if -2.79999999999999983e128 < b < -5.2000000000000003e36

   1. Initial program 49.3

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

   2. Simplified 46.8

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

   3. Taylor expanded in x-scale around 0 44.5

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around inf 40.1

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

   5. Taylor expanded in x-scale around 0 38.3

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

   6. Simplified 37.8

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

   7. Taylor expanded in angle around 0 37.6

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

   if -1.2e22 < b < -9.0000000000000001e-286

   1. Initial program 53.1

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

   2. Simplified 51.5

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

   3. Taylor expanded in x-scale around 0 48.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around inf 32.4

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

   5. Taylor expanded in x-scale around 0 28.8

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

   6. Simplified 26.8

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

   7. Applied egg-rr 26.7

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

   if -9.0000000000000001e-286 < b < 6.7000000000000001e-12

   1. Initial program 54.5

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

   2. Simplified 52.4

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

   3. Taylor expanded in x-scale around 0 49.6

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around inf 32.3

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

   5. Taylor expanded in x-scale around 0 27.6

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

   6. Simplified 26.1

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

   7. Applied egg-rr 26.0

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

   if 6.5999999999999996e46 < b < 2.49999999999999992e78

   1. Initial program 50.8

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

   2. Simplified 48.1

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

   3. Taylor expanded in x-scale around 0 40.7

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around inf 40.2

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

   5. Taylor expanded in x-scale around 0 37.0

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

   6. Simplified 36.6

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

   7. Applied egg-rr 36.6

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

   if 2.49999999999999992e78 < b

   1. Initial program 59.9

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

   2. Simplified 58.5

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

   3. Taylor expanded in x-scale around 0 58.6

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\left(-0.5 \cdot \frac{y-scale \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right) + \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4} + {a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({x-scale}^{2} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}\right)}{\pi} \]

   4. Taylor expanded in a around 0 29.6

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \color{blue}{\left(2 \cdot \frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}\right)\right)}{\pi} \]
3. Recombined 6 regimes into one program.

4. Final simplification 28.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+128}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale}{\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(x-scale \cdot x-scale\right)}{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+22}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale}{\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(x-scale \cdot x-scale\right)}{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-286}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{\frac{y-scale}{\cos \left({\left(\sqrt[3]{\left(0.005555555555555556 \cdot angle\right) \cdot \pi}\right)}^{3}\right)}}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-12}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{x-scale \cdot \frac{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{y-scale}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+46}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale}{\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(x-scale \cdot x-scale\right)}{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+78}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{\frac{y-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \left(-0.5 \cdot \left(2 \cdot \frac{y-scale \cdot \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)\right)}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(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)))