a from scale-rotated-ellipse

Average Accuracy: 0.9% → 33.9%

Time: 1.8min

Precision: binary64

Cost: 98436

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := a \cdot t_2\\ t_4 := b \cdot t_1\\ \mathbf{if}\;x-scale \leq -1.56 \cdot 10^{-108}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(\left(\sqrt[3]{{t_3}^{2} + {t_4}^{2}} \cdot \sqrt[3]{\mathsf{hypot}\left(t_4, t_3\right)}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 5.7 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {t_2}^{2}, b \cdot \left(b \cdot {t_1}^{2}\right)\right)}\right)\right)\\ \end{array} \]

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 (* PI (* angle 0.005555555555555556)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3 (* a t_2))
        (t_4 (* b t_1)))
   (if (<= x-scale -1.56e-108)
     (*
      (* (sqrt 8.0) (* x-scale (sqrt 2.0)))
      (*
       (* (cbrt (+ (pow t_3 2.0) (pow t_4 2.0))) (cbrt (hypot t_4 t_3)))
       -0.25))
     (if (<= x-scale 5.7e+78)
       (* 0.25 (fabs (* (* (sqrt 8.0) b) (* (sqrt 2.0) y-scale))))
       (*
        (sqrt 2.0)
        (*
         0.25
         (*
          (* x-scale (sqrt 8.0))
          (sqrt (fma (* a a) (pow t_2 2.0) (* b (* b (pow t_1 2.0))))))))))))

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 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = a * t_2;
	double t_4 = b * t_1;
	double tmp;
	if (x_45_scale <= -1.56e-108) {
		tmp = (sqrt(8.0) * (x_45_scale * sqrt(2.0))) * ((cbrt((pow(t_3, 2.0) + pow(t_4, 2.0))) * cbrt(hypot(t_4, t_3))) * -0.25);
	} else if (x_45_scale <= 5.7e+78) {
		tmp = 0.25 * fabs(((sqrt(8.0) * b) * (sqrt(2.0) * y_45_scale)));
	} else {
		tmp = sqrt(2.0) * (0.25 * ((x_45_scale * sqrt(8.0)) * sqrt(fma((a * a), pow(t_2, 2.0), (b * (b * pow(t_1, 2.0)))))));
	}
	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(pi * Float64(angle * 0.005555555555555556))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(a * t_2)
	t_4 = Float64(b * t_1)
	tmp = 0.0
	if (x_45_scale <= -1.56e-108)
		tmp = Float64(Float64(sqrt(8.0) * Float64(x_45_scale * sqrt(2.0))) * Float64(Float64(cbrt(Float64((t_3 ^ 2.0) + (t_4 ^ 2.0))) * cbrt(hypot(t_4, t_3))) * -0.25));
	elseif (x_45_scale <= 5.7e+78)
		tmp = Float64(0.25 * abs(Float64(Float64(sqrt(8.0) * b) * Float64(sqrt(2.0) * y_45_scale))));
	else
		tmp = Float64(sqrt(2.0) * Float64(0.25 * Float64(Float64(x_45_scale * sqrt(8.0)) * sqrt(fma(Float64(a * a), (t_2 ^ 2.0), Float64(b * Float64(b * (t_1 ^ 2.0))))))));
	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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(a * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(b * t$95$1), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.56e-108], N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Power[t$95$3, 2.0], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sqrt[t$95$4 ^ 2 + t$95$3 ^ 2], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 5.7e+78], N[(0.25 * N[Abs[N[(N[(N[Sqrt[8.0], $MachinePrecision] * b), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[(N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision] + N[(b * N[(b * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.56000000000000009e-108

   1. Initial program 1.2%

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

   2. Simplified 1.8%

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

      [Start] 1.2

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

   3. Taylor expanded in x-scale around -inf 5.4%

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

   4. Simplified 5.4%

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

      [Start] 5.4	\[ -0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 5.4	\[ -0.25 \cdot \left(\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      *-commutative [<=] 5.4	\[ -0.25 \cdot \left(\left(\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [<=] 5.4	\[ -0.25 \cdot \left(\color{blue}{\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 5.4	\[ \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}} \]
      associate-*r* [=>] 5.4	\[ \left(-0.25 \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      *-commutative [=>] 5.4	\[ \left(-0.25 \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      associate-*r* [<=] 5.4	\[ \left(-0.25 \cdot \color{blue}{\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      *-commutative [=>] 5.4	\[ \left(-0.25 \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot x-scale\right)}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]

   5. Taylor expanded in y-scale around 0 25.9%

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

   6. Simplified 28.0%

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

      [Start] 25.9	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 25.9	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot -0.25} \]
      associate-*l* [=>] 25.9	\[ \color{blue}{\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right)} \]
      associate-*r* [=>] 25.9	\[ \color{blue}{\left(\left(\sqrt{2} \cdot x-scale\right) \cdot \sqrt{8}\right)} \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right) \]
      *-commutative [=>] 25.9	\[ \color{blue}{\left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right)} \cdot \left(\sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right) \]

   7. Applied egg-rr 30.0%

      \[\leadsto \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\sqrt[3]{{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt[3]{\sqrt{{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}\right)} \cdot -0.25\right) \]

   8. Simplified 30.0%

      \[\leadsto \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\sqrt[3]{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt[3]{\mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)} \cdot -0.25\right) \]
      Proof

      [Start] 30.0	\[ \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\left(\sqrt[3]{{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt[3]{\sqrt{{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}\right) \cdot -0.25\right) \]
      +-commutative [=>] 30.0	\[ \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}\right) \cdot -0.25\right) \]
      +-commutative [=>] 30.0	\[ \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot x-scale\right)\right) \cdot \left(\left(\sqrt[3]{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt[3]{\sqrt{\color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}}\right) \cdot -0.25\right) \]

   if -1.56000000000000009e-108 < x-scale < 5.69999999999999986e78

   1. Initial program 0.8%

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

   2. Taylor expanded in angle around 0 19.9%

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

   3. Applied egg-rr 35.1%

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

   if 5.69999999999999986e78 < x-scale

   1. Initial program 0.8%

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

   2. Simplified 1.2%

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

      [Start] 0.8

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

   3. Taylor expanded in x-scale around -inf 7.9%

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

   4. Simplified 8.0%

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

      [Start] 7.9	\[ -0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 7.9	\[ -0.25 \cdot \left(\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      *-commutative [<=] 7.9	\[ -0.25 \cdot \left(\left(\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [<=] 7.9	\[ -0.25 \cdot \left(\color{blue}{\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 7.9	\[ \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}} \]
      associate-*r* [=>] 7.9	\[ \left(-0.25 \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      *-commutative [=>] 7.9	\[ \left(-0.25 \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      associate-*r* [<=] 7.9	\[ \left(-0.25 \cdot \color{blue}{\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      *-commutative [=>] 7.9	\[ \left(-0.25 \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot x-scale\right)}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]

   5. Taylor expanded in y-scale around -inf 33.9%

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

   6. Simplified 37.7%

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

      [Start] 33.9	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 33.9	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot 0.25} \]
      associate-*l* [=>] 33.9	\[ \color{blue}{\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)} \cdot 0.25 \]
      associate-*l* [=>] 33.9	\[ \color{blue}{\sqrt{2} \cdot \left(\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot 0.25\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.56 \cdot 10^{-108}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(\left(\sqrt[3]{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \cdot \sqrt[3]{\mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 5.7 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}, b \cdot \left(b \cdot {\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	32.6%
Cost	65864

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_2 := x-scale \cdot \sqrt{8}\\ \mathbf{if}\;x-scale \leq -2.6 \cdot 10^{-108}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\sin t_1}^{2}, a \cdot \left(a \cdot {\cos t_1}^{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;x-scale \leq 5.7 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(t_2 \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos t_0}^{2}, b \cdot \left(b \cdot {\sin t_0}^{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	33.3%
Cost	65864

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sqrt{\mathsf{fma}\left(a \cdot a, {\cos t_0}^{2}, b \cdot \left(b \cdot {\sin t_0}^{2}\right)\right)}\\ \mathbf{if}\;x-scale \leq -2.6 \cdot 10^{-108}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(-0.25 \cdot t_1\right)\\ \mathbf{elif}\;x-scale \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot t_1\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	32.1%
Cost	65732

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := {\cos t_0}^{2}\\ t_2 := {\sin t_0}^{2}\\ t_3 := x-scale \cdot \sqrt{8}\\ \mathbf{if}\;x-scale \leq -1.1 \cdot 10^{-112}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(t_3 \cdot \sqrt{\mathsf{fma}\left(b \cdot b, t_2, a \cdot \left(a \cdot t_1\right)\right)}\right)\right)\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot t_3\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(b \cdot b, t_2, \left(a \cdot a\right) \cdot t_1\right)}\\ \end{array} \]

Alternative 4
Accuracy	31.0%
Cost	65668

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;x-scale \leq -1.62 \cdot 10^{+36}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(-0.25 \cdot \sqrt[3]{{\left({\left(a \cdot \cos t_1\right)}^{2} + {\left(b \cdot \sin t_1\right)}^{2}\right)}^{1.5}}\right)\\ \mathbf{elif}\;x-scale \leq -1.2 \cdot 10^{-104}:\\ \;\;\;\;\left(4 \cdot \left|x-scale\right|\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{+79}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(b \cdot b, {\sin t_0}^{2}, \left(a \cdot a\right) \cdot {\cos t_0}^{2}\right)}\\ \end{array} \]

Alternative 5
Accuracy	29.9%
Cost	20304

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -9.5 \cdot 10^{-219}:\\ \;\;\;\;0.25 \cdot \left|y-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot b\right)\right)\right|\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{-279}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 4.6 \cdot 10^{-207}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.9 \cdot 10^{-153}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left|\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot y-scale\right)\right)\right|\\ \end{array} \]

Alternative 6
Accuracy	30.0%
Cost	20304

\[\begin{array}{l} t_0 := 0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{if}\;y-scale \leq -1.2 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 1.8 \cdot 10^{-207}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.5 \cdot 10^{-154}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	29.9%
Cost	20176

\[\begin{array}{l} t_0 := \left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(b \cdot y-scale\right)\right)\right|\\ \mathbf{if}\;y-scale \leq -4 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 5.5 \cdot 10^{-281}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{-207}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{-153}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	29.9%
Cost	20176

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.55 \cdot 10^{-217}:\\ \;\;\;\;0.25 \cdot \left|y-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot b\right)\right)\right|\\ \mathbf{elif}\;y-scale \leq 6.6 \cdot 10^{-280}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 4.1 \cdot 10^{-207}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(b \cdot y-scale\right)\right)\right|\\ \end{array} \]

Alternative 9
Accuracy	31.8%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;a \leq 3.75 \cdot 10^{+31}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{2} \cdot y-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left|x-scale\right|\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	21.3%
Cost	13964

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -5.9 \cdot 10^{-189}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{-279}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 3.6 \cdot 10^{-209}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.4 \cdot 10^{-82}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 11
Accuracy	19.3%
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-280}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-97}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+58}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	19.1%
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot a\right)\right)\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+67}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	19.2%
Cost	13904

\[\begin{array}{l} t_0 := x-scale \cdot \sqrt{2}\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{+75}:\\ \;\;\;\;\left(\sqrt{8} \cdot t_0\right) \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-97}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+69}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(t_0 \cdot \left(\sqrt{8} \cdot a\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	21.1%
Cost	13904

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\\ \mathbf{if}\;y-scale \leq -1.02 \cdot 10^{-187}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;y-scale \leq 2.3 \cdot 10^{-282}:\\ \;\;\;\;t_0 \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{-207}:\\ \;\;\;\;t_0 \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;y-scale \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 15
Accuracy	16.3%
Cost	13452

\[\begin{array}{l} \mathbf{if}\;angle \leq 2.5 \cdot 10^{-273}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;angle \leq 2.8 \cdot 10^{-158}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;angle \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(b \cdot y-scale\right)}^{2}}\\ \end{array} \]

Alternative 16
Accuracy	15.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-282} \lor \neg \left(a \leq 1.02 \cdot 10^{-97}\right):\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \end{array} \]

Alternative 17
Accuracy	15.9%
Cost	192

\[b \cdot y-scale \]

Reproduce

herbie shell --seed 2023122 
(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))))