a from scale-rotated-ellipse

Average Accuracy: 0.9% → 40.4%

Time: 2.0min

Precision: binary64

Cost: 46608

Math

\[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

↓

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_2 := \mathsf{hypot}\left(a \cdot \cos t_1, b \cdot \sin t_1\right)\\ t_3 := t_2 \cdot \left(-0.25 \cdot t_0\right)\\ \mathbf{if}\;x-scale \leq -6.2 \cdot 10^{+90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq -1.12 \cdot 10^{+42}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{b \cdot 4} \cdot \sqrt[3]{y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 3.2 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot 0.25\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (+
      (+
       (/
        (/
         (+
          (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))
      (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)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (sqrt 8.0) (* x-scale (sqrt 2.0))))
        (t_1 (* angle (* PI 0.005555555555555556)))
        (t_2 (hypot (* a (cos t_1)) (* b (sin t_1))))
        (t_3 (* t_2 (* -0.25 t_0))))
   (if (<= x-scale -6.2e+90)
     t_3
     (if (<= x-scale -1.12e+42)
       (* 0.25 (pow (pow (* (cbrt (* b 4.0)) (cbrt y-scale)) 2.0) 1.5))
       (if (<= x-scale -1.6e-162)
         t_3
         (if (<= x-scale 3.2e-99)
           (* 0.25 (pow (pow (cbrt (* (* b 4.0) y-scale)) 2.0) 1.5))
           (* t_0 (* t_2 0.25))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) + 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)))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt(8.0) * (x_45_scale * sqrt(2.0));
	double t_1 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_2 = hypot((a * cos(t_1)), (b * sin(t_1)));
	double t_3 = t_2 * (-0.25 * t_0);
	double tmp;
	if (x_45_scale <= -6.2e+90) {
		tmp = t_3;
	} else if (x_45_scale <= -1.12e+42) {
		tmp = 0.25 * pow(pow((cbrt((b * 4.0)) * cbrt(y_45_scale)), 2.0), 1.5);
	} else if (x_45_scale <= -1.6e-162) {
		tmp = t_3;
	} else if (x_45_scale <= 3.2e-99) {
		tmp = 0.25 * pow(pow(cbrt(((b * 4.0) * y_45_scale)), 2.0), 1.5);
	} else {
		tmp = t_0 * (t_2 * 0.25);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -Math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) + 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)))))) / ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0));
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sqrt(8.0) * (x_45_scale * Math.sqrt(2.0));
	double t_1 = angle * (Math.PI * 0.005555555555555556);
	double t_2 = Math.hypot((a * Math.cos(t_1)), (b * Math.sin(t_1)));
	double t_3 = t_2 * (-0.25 * t_0);
	double tmp;
	if (x_45_scale <= -6.2e+90) {
		tmp = t_3;
	} else if (x_45_scale <= -1.12e+42) {
		tmp = 0.25 * Math.pow(Math.pow((Math.cbrt((b * 4.0)) * Math.cbrt(y_45_scale)), 2.0), 1.5);
	} else if (x_45_scale <= -1.6e-162) {
		tmp = t_3;
	} else if (x_45_scale <= 3.2e-99) {
		tmp = 0.25 * Math.pow(Math.pow(Math.cbrt(((b * 4.0) * y_45_scale)), 2.0), 1.5);
	} else {
		tmp = t_0 * (t_2 * 0.25);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * 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)) + 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(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(sqrt(8.0) * Float64(x_45_scale * sqrt(2.0)))
	t_1 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_2 = hypot(Float64(a * cos(t_1)), Float64(b * sin(t_1)))
	t_3 = Float64(t_2 * Float64(-0.25 * t_0))
	tmp = 0.0
	if (x_45_scale <= -6.2e+90)
		tmp = t_3;
	elseif (x_45_scale <= -1.12e+42)
		tmp = Float64(0.25 * ((Float64(cbrt(Float64(b * 4.0)) * cbrt(y_45_scale)) ^ 2.0) ^ 1.5));
	elseif (x_45_scale <= -1.6e-162)
		tmp = t_3;
	elseif (x_45_scale <= 3.2e-99)
		tmp = Float64(0.25 * ((cbrt(Float64(Float64(b * 4.0) * y_45_scale)) ^ 2.0) ^ 1.5));
	else
		tmp = Float64(t_0 * Float64(t_2 * 0.25));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[((-N[Sqrt[N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(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] + 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]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -6.2e+90], t$95$3, If[LessEqual[x$45$scale, -1.12e+42], N[(0.25 * N[Power[N[Power[N[(N[Power[N[(b * 4.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[y$45$scale, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -1.6e-162], t$95$3, If[LessEqual[x$45$scale, 3.2e-99], N[(0.25 * N[Power[N[Power[N[Power[N[(N[(b * 4.0), $MachinePrecision] * y$45$scale), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x-scale < -6.19999999999999977e90 or -1.12e42 < x-scale < -1.59999999999999988e-162

   1. Initial program 1.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Simplified 1.7%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(\left(b \cdot a\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{x-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 1.0

      \[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Taylor expanded in x-scale around -inf 3.8%

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\frac{\sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)}\right) \]

   4. Simplified 3.8%

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)} \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right)}\right) \]
      Proof

      [Start] 3.8

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\frac{\sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]

      *-commutative [=>] 3.8

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right)}\right) \]

   5. Taylor expanded in y-scale around 0 25.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)} \]

   6. Simplified 40.8%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-0.25 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right)\right)} \]
      Proof

      [Start] 25.0	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      associate-*r* [=>] 25.0	\[ \color{blue}{\left(-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} \]
      *-commutative [=>] 25.0	\[ \color{blue}{\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \left(-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)} \]

   if -6.19999999999999977e90 < x-scale < -1.12e42

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Simplified 2.4%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(\left(b \cdot a\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{x-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 1.9

      \[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Taylor expanded in angle around 0 9.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)} \]

   4. Simplified 9.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)} \]
      Proof

      [Start] 9.9	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 9.9	\[ 0.25 \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 9.9	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{8} \cdot b\right)}\right) \]

   5. Applied egg-rr 4.9%

      \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right) \cdot 0.3333333333333333}} \]
      Proof

      [Start] 9.9	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \]
      add-cbrt-cube [=>] 5.3	\[ 0.25 \cdot \color{blue}{\sqrt[3]{\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)}} \]
      pow1/3 [=>] 4.8	\[ 0.25 \cdot \color{blue}{{\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right)}^{0.3333333333333333}} \]
      pow-to-exp [=>] 4.9	\[ 0.25 \cdot \color{blue}{e^{\log \left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   6. Applied egg-rr 20.2%

      \[\leadsto 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot 4\right)}\right)}^{2}\right)}^{1.5}} \]
      Proof

      [Start] 4.9	\[ 0.25 \cdot e^{\log \left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right) \cdot 0.3333333333333333} \]
      exp-to-pow [=>] 4.8	\[ 0.25 \cdot \color{blue}{{\left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      pow1/3 [<=] 5.4	\[ 0.25 \cdot \color{blue}{\sqrt[3]{{\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}}} \]
      rem-cbrt-cube [=>] 10.0	\[ 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)} \]
      add-cube-cbrt [=>] 9.8	\[ 0.25 \cdot \color{blue}{\left(\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)} \cdot \sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right) \cdot \sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)} \]
      pow3 [=>] 9.8	\[ 0.25 \cdot \color{blue}{{\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{3}} \]
      metadata-eval [<=] 9.8	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\color{blue}{\left(2 \cdot 1.5\right)}} \]
      metadata-eval [<=] 9.8	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)} \]
      metadata-eval [<=] 9.8	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\color{blue}{\sqrt{4}} \cdot \frac{3}{2}\right)} \]
      metadata-eval [<=] 9.8	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{\color{blue}{\sqrt{16}}} \cdot \frac{3}{2}\right)} \]
      unpow1/2 [<=] 9.8	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{\color{blue}{{16}^{0.5}}} \cdot \frac{3}{2}\right)} \]
      pow-pow [<=] 20.2	\[ 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{{16}^{0.5}}\right)}\right)}^{\left(\frac{3}{2}\right)}} \]

   7. Simplified 20.2%

      \[\leadsto 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{y-scale \cdot \left(b \cdot 4\right)}\right)}^{2}\right)}^{1.5}} \]
      Proof

      [Start] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot 4\right)}\right)}^{2}\right)}^{1.5} \]
      *-commutative [=>] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{\color{blue}{\left(y-scale \cdot 4\right) \cdot b}}\right)}^{2}\right)}^{1.5} \]
      associate-*r* [<=] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{\color{blue}{y-scale \cdot \left(4 \cdot b\right)}}\right)}^{2}\right)}^{1.5} \]
      *-commutative [=>] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{y-scale \cdot \color{blue}{\left(b \cdot 4\right)}}\right)}^{2}\right)}^{1.5} \]

   8. Applied egg-rr 20.1%

      \[\leadsto 0.25 \cdot {\left({\color{blue}{\left(\sqrt[3]{b \cdot 4} \cdot \sqrt[3]{y-scale}\right)}}^{2}\right)}^{1.5} \]
      Proof

      [Start] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{y-scale \cdot \left(b \cdot 4\right)}\right)}^{2}\right)}^{1.5} \]
      *-commutative [=>] 20.2	\[ 0.25 \cdot {\left({\left(\sqrt[3]{\color{blue}{\left(b \cdot 4\right) \cdot y-scale}}\right)}^{2}\right)}^{1.5} \]
      cbrt-prod [=>] 20.1	\[ 0.25 \cdot {\left({\color{blue}{\left(\sqrt[3]{b \cdot 4} \cdot \sqrt[3]{y-scale}\right)}}^{2}\right)}^{1.5} \]

   if -1.59999999999999988e-162 < x-scale < 3.2000000000000001e-99

   1. Initial program 0.3%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Simplified 0.1%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(\left(b \cdot a\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{x-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 0.3

      \[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Taylor expanded in angle around 0 22.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)} \]

   4. Simplified 22.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)} \]
      Proof

      [Start] 22.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 22.5	\[ 0.25 \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 22.5	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{8} \cdot b\right)}\right) \]

   5. Applied egg-rr 12.3%

      \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right) \cdot 0.3333333333333333}} \]
      Proof

      [Start] 22.5	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \]
      add-cbrt-cube [=>] 12.9	\[ 0.25 \cdot \color{blue}{\sqrt[3]{\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)}} \]
      pow1/3 [=>] 12.3	\[ 0.25 \cdot \color{blue}{{\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right)}^{0.3333333333333333}} \]
      pow-to-exp [=>] 12.3	\[ 0.25 \cdot \color{blue}{e^{\log \left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   6. Applied egg-rr 38.7%

      \[\leadsto 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot 4\right)}\right)}^{2}\right)}^{1.5}} \]
      Proof

      [Start] 12.3	\[ 0.25 \cdot e^{\log \left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right) \cdot 0.3333333333333333} \]
      exp-to-pow [=>] 12.3	\[ 0.25 \cdot \color{blue}{{\left({\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      pow1/3 [<=] 13.0	\[ 0.25 \cdot \color{blue}{\sqrt[3]{{\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{3}}} \]
      rem-cbrt-cube [=>] 22.7	\[ 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)} \]
      add-cube-cbrt [=>] 22.4	\[ 0.25 \cdot \color{blue}{\left(\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)} \cdot \sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right) \cdot \sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)} \]
      pow3 [=>] 22.4	\[ 0.25 \cdot \color{blue}{{\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{3}} \]
      metadata-eval [<=] 22.4	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\color{blue}{\left(2 \cdot 1.5\right)}} \]
      metadata-eval [<=] 22.4	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)} \]
      metadata-eval [<=] 22.4	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\color{blue}{\sqrt{4}} \cdot \frac{3}{2}\right)} \]
      metadata-eval [<=] 22.4	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{\color{blue}{\sqrt{16}}} \cdot \frac{3}{2}\right)} \]
      unpow1/2 [<=] 22.4	\[ 0.25 \cdot {\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{\color{blue}{{16}^{0.5}}} \cdot \frac{3}{2}\right)} \]
      pow-pow [<=] 38.7	\[ 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot {16}^{0.5}\right)}\right)}^{\left(\sqrt{{16}^{0.5}}\right)}\right)}^{\left(\frac{3}{2}\right)}} \]

   7. Simplified 38.7%

      \[\leadsto 0.25 \cdot \color{blue}{{\left({\left(\sqrt[3]{y-scale \cdot \left(b \cdot 4\right)}\right)}^{2}\right)}^{1.5}} \]
      Proof

      [Start] 38.7	\[ 0.25 \cdot {\left({\left(\sqrt[3]{b \cdot \left(y-scale \cdot 4\right)}\right)}^{2}\right)}^{1.5} \]
      *-commutative [=>] 38.7	\[ 0.25 \cdot {\left({\left(\sqrt[3]{\color{blue}{\left(y-scale \cdot 4\right) \cdot b}}\right)}^{2}\right)}^{1.5} \]
      associate-*r* [<=] 38.7	\[ 0.25 \cdot {\left({\left(\sqrt[3]{\color{blue}{y-scale \cdot \left(4 \cdot b\right)}}\right)}^{2}\right)}^{1.5} \]
      *-commutative [=>] 38.7	\[ 0.25 \cdot {\left({\left(\sqrt[3]{y-scale \cdot \color{blue}{\left(b \cdot 4\right)}}\right)}^{2}\right)}^{1.5} \]

   if 3.2000000000000001e-99 < x-scale

   1. Initial program 1.3%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Simplified 2.1%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(\left(b \cdot a\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{x-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 1.3

      \[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Taylor expanded in x-scale around -inf 4.2%

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\frac{\sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)}\right) \]

   4. Simplified 4.2%

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)} \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right)}\right) \]
      Proof

      [Start] 4.2

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\frac{\sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]

      *-commutative [=>] 4.2

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right)}\right) \]

   5. Taylor expanded in y-scale around -inf 26.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)} \]

   6. Simplified 44.8%

      \[\leadsto \color{blue}{\left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot 0.25\right)} \]
      Proof

      [Start] 26.7	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 26.7	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot 0.25} \]
      associate-*l* [=>] 26.8	\[ \color{blue}{\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot 0.25\right)} \]
      associate-*r* [=>] 26.8	\[ \color{blue}{\left(\left(\sqrt{2} \cdot x-scale\right) \cdot \sqrt{8}\right)} \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot 0.25\right) \]
      *-commutative [=>] 26.8	\[ \color{blue}{\left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right)} \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot 0.25\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -6.2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-0.25 \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.12 \cdot 10^{+42}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{b \cdot 4} \cdot \sqrt[3]{y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-0.25 \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.2 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	35.7%
Cost	46344

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{+136}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \left(\left(t_1 \cdot \sqrt{8}\right) \cdot \left(a \cdot \sqrt{2}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot t_1, b \cdot \sin t_0\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 2
Accuracy	27.6%
Cost	27024

\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(x-scale \cdot \sqrt{2}\right) \cdot t_0\\ \mathbf{if}\;x-scale \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.8 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 10^{+136}:\\ \;\;\;\;-0.25 \cdot t_1\\ \mathbf{elif}\;x-scale \leq 5.8 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	27.3%
Cost	27024

\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(x-scale \cdot \sqrt{2}\right) \cdot t_0\\ \mathbf{if}\;x-scale \leq -2.02 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq 7 \cdot 10^{-100}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 7.2 \cdot 10^{+135}:\\ \;\;\;\;-0.25 \cdot t_1\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{+252}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	27.4%
Cost	27024

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(x-scale \cdot \sqrt{2}\right) \cdot \left(a \cdot \sqrt{8}\right)\\ \mathbf{if}\;x-scale \leq -1.35 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos t_0\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.8 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 10^{+137}:\\ \;\;\;\;-0.25 \cdot t_1\\ \mathbf{elif}\;x-scale \leq 2.4 \cdot 10^{+252}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot \sin t_0\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	27.3%
Cost	27024

\[\begin{array}{l} t_0 := \left(x-scale \cdot \sqrt{2}\right) \cdot \left(a \cdot \sqrt{8}\right)\\ \mathbf{if}\;x-scale \leq -1.4 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\left(a \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.65 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{+135}:\\ \;\;\;\;-0.25 \cdot t_0\\ \mathbf{elif}\;x-scale \leq 1.16 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	27.4%
Cost	27024

\[\begin{array}{l} t_0 := \left(x-scale \cdot \sqrt{2}\right) \cdot \left(a \cdot \sqrt{8}\right)\\ \mathbf{if}\;x-scale \leq -1.4 \cdot 10^{+136}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \left(\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sqrt{8}\right) \cdot \left(a \cdot \sqrt{2}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.8 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{+136}:\\ \;\;\;\;-0.25 \cdot t_0\\ \mathbf{elif}\;x-scale \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;0.25 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	20.6%
Cost	20304

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{{\left(b \cdot \left(y-scale \cdot {16}^{0.5}\right)\right)}^{2}}\\ t_1 := a \cdot \sqrt{8}\\ t_2 := \left(x-scale \cdot \sqrt{2}\right) \cdot t_1\\ t_3 := -0.25 \cdot t_2\\ \mathbf{if}\;x-scale \leq -1.85 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right)\\ \mathbf{elif}\;x-scale \leq -1650000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -7 \cdot 10^{-84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 3.8 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 6.5 \cdot 10^{+136} \lor \neg \left(x-scale \leq 2.25 \cdot 10^{+245}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot t_2\\ \end{array} \]

Alternative 8
Accuracy	27.7%
Cost	20040

\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(x-scale \cdot \sqrt{2}\right) \cdot t_0\\ \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.4 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot {\left({\left(\sqrt[3]{\left(b \cdot 4\right) \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 9.6 \cdot 10^{+135} \lor \neg \left(x-scale \leq 4.6 \cdot 10^{+250}\right):\\ \;\;\;\;-0.25 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot t_1\\ \end{array} \]

Alternative 9
Accuracy	19.6%
Cost	14169

\[\begin{array}{l} t_0 := b \cdot \left(4 \cdot y-scale\right)\\ t_1 := a \cdot \sqrt{8}\\ t_2 := -0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot t_1\right)\\ t_3 := 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right)\\ \mathbf{if}\;x-scale \leq -1.42 \cdot 10^{+136}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq -1520000000:\\ \;\;\;\;0.25 \cdot \sqrt{\left(4 \cdot y-scale\right) \cdot \left(b \cdot t_0\right)}\\ \mathbf{elif}\;x-scale \leq -2.7 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 2.5 \cdot 10^{-100}:\\ \;\;\;\;0.25 \cdot \sqrt{b \cdot \left(\left(4 \cdot y-scale\right) \cdot t_0\right)}\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{+135} \lor \neg \left(x-scale \leq 1.25 \cdot 10^{+252}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Accuracy	19.5%
Cost	14169

\[\begin{array}{l} t_0 := b \cdot \left(4 \cdot y-scale\right)\\ t_1 := a \cdot \sqrt{8}\\ t_2 := \left(x-scale \cdot \sqrt{2}\right) \cdot t_1\\ t_3 := -0.25 \cdot t_2\\ \mathbf{if}\;x-scale \leq -1.42 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right)\\ \mathbf{elif}\;x-scale \leq -1150000000:\\ \;\;\;\;0.25 \cdot \sqrt{\left(4 \cdot y-scale\right) \cdot \left(b \cdot t_0\right)}\\ \mathbf{elif}\;x-scale \leq -9.2 \cdot 10^{-81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 3 \cdot 10^{-100}:\\ \;\;\;\;0.25 \cdot \sqrt{b \cdot \left(\left(4 \cdot y-scale\right) \cdot t_0\right)}\\ \mathbf{elif}\;x-scale \leq 5.4 \cdot 10^{+135} \lor \neg \left(x-scale \leq 7 \cdot 10^{+244}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot t_2\\ \end{array} \]

Alternative 11
Accuracy	19.8%
Cost	13905

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(a \cdot \sqrt{8}\right)\right)\\ t_1 := b \cdot \left(4 \cdot y-scale\right)\\ \mathbf{if}\;x-scale \leq -3.3 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -265000000:\\ \;\;\;\;0.25 \cdot \sqrt{\left(4 \cdot y-scale\right) \cdot \left(b \cdot t_1\right)}\\ \mathbf{elif}\;x-scale \leq -9.5 \cdot 10^{-81} \lor \neg \left(x-scale \leq 2.9 \cdot 10^{-100}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{b \cdot \left(\left(4 \cdot y-scale\right) \cdot t_1\right)}\\ \end{array} \]

Alternative 12
Accuracy	16.1%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.2 \cdot 10^{-212}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;angle \leq -8.2 \cdot 10^{-294}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot 4\right) \cdot y-scale\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\left(4 \cdot y-scale\right) \cdot \left(b \cdot \left(b \cdot \left(4 \cdot y-scale\right)\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	16.0%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;angle \leq -1 \cdot 10^{-220}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;angle \leq -9 \cdot 10^{-264}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot 4\right) \cdot y-scale\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{b \cdot \left(\left(4 \cdot y-scale\right) \cdot \left(b \cdot \left(4 \cdot y-scale\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	15.7%
Cost	978

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.7 \cdot 10^{-218} \lor \neg \left(angle \leq -4.7 \cdot 10^{-284} \lor \neg \left(angle \leq 3.2 \cdot 10^{-229}\right) \land angle \leq 5.8 \cdot 10^{-156}\right):\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot 4\right) \cdot y-scale\right)\\ \end{array} \]

Alternative 15
Accuracy	15.5%
Cost	448

\[0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right) \]

Reproduce

herbie shell --seed 2023146 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (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)) (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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))