a from scale-rotated-ellipse

Average Error: 63.4 → 46.9

Time: 1.8min

Precision: binary64

Cost: 72200

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\\ t_3 := \cos t_0\\ \mathbf{if}\;y-scale \leq -1.15 \cdot 10^{+27}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot a, {t_1}^{2}, {t_3}^{2} \cdot \left(b \cdot b\right)\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 8.2 \cdot 10^{-49}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{{\left(t_3 \cdot b\right)}^{2} + {\left(a \cdot t_1\right)}^{2}}}\right)}^{3} \cdot \left(t_2 \cdot 0.25\right)\\ \end{array} \]

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 (* PI (* angle 0.005555555555555556)))
        (t_1 (sin t_0))
        (t_2 (* (sqrt 8.0) (* y-scale (sqrt 2.0))))
        (t_3 (cos t_0)))
   (if (<= y-scale -1.15e+27)
     (*
      t_2
      (* (sqrt (fma (* a a) (pow t_1 2.0) (* (pow t_3 2.0) (* b b)))) -0.25))
     (if (<= y-scale 8.2e-49)
       (*
        0.25
        (sqrt
         (*
          2.0
          (pow
           (*
            a
            (*
             (* (sqrt 8.0) x-scale)
             (cos (* angle (* PI 0.005555555555555556)))))
           2.0))))
       (*
        (pow (cbrt (sqrt (+ (pow (* t_3 b) 2.0) (pow (* a t_1) 2.0)))) 3.0)
        (* 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 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = sin(t_0);
	double t_2 = sqrt(8.0) * (y_45_scale * sqrt(2.0));
	double t_3 = cos(t_0);
	double tmp;
	if (y_45_scale <= -1.15e+27) {
		tmp = t_2 * (sqrt(fma((a * a), pow(t_1, 2.0), (pow(t_3, 2.0) * (b * b)))) * -0.25);
	} else if (y_45_scale <= 8.2e-49) {
		tmp = 0.25 * sqrt((2.0 * pow((a * ((sqrt(8.0) * x_45_scale) * cos((angle * (((double) M_PI) * 0.005555555555555556))))), 2.0)));
	} else {
		tmp = pow(cbrt(sqrt((pow((t_3 * b), 2.0) + pow((a * t_1), 2.0)))), 3.0) * (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(pi * Float64(angle * 0.005555555555555556))
	t_1 = sin(t_0)
	t_2 = Float64(sqrt(8.0) * Float64(y_45_scale * sqrt(2.0)))
	t_3 = cos(t_0)
	tmp = 0.0
	if (y_45_scale <= -1.15e+27)
		tmp = Float64(t_2 * Float64(sqrt(fma(Float64(a * a), (t_1 ^ 2.0), Float64((t_3 ^ 2.0) * Float64(b * b)))) * -0.25));
	elseif (y_45_scale <= 8.2e-49)
		tmp = Float64(0.25 * sqrt(Float64(2.0 * (Float64(a * Float64(Float64(sqrt(8.0) * x_45_scale) * cos(Float64(angle * Float64(pi * 0.005555555555555556))))) ^ 2.0))));
	else
		tmp = Float64((cbrt(sqrt(Float64((Float64(t_3 * b) ^ 2.0) + (Float64(a * t_1) ^ 2.0)))) ^ 3.0) * 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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[8.0], $MachinePrecision] * N[(y$45$scale * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale, -1.15e+27], N[(t$95$2 * N[(N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 8.2e-49], N[(0.25 * N[Sqrt[N[(2.0 * N[Power[N[(a * N[(N[(N[Sqrt[8.0], $MachinePrecision] * x$45$scale), $MachinePrecision] * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[Sqrt[N[(N[Power[N[(t$95$3 * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.15e27

   1. Initial program 63.4

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

   2. Taylor expanded in y-scale around -inf 59.6

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

   3. Simplified 59.6

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

      [Start] 59.6	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      associate-*r* [=>] 59.6	\[ \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}} \]
      associate-*r* [=>] 59.6	\[ \left(-0.25 \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \]
      *-commutative [<=] 59.6	\[ \left(-0.25 \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \]
      distribute-lft-out [=>] 59.6	\[ \left(-0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}} \]
      +-commutative [=>] 59.6	\[ \left(-0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}} \]

   4. Taylor expanded in x-scale around 0 43.5

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

   5. Simplified 43.5

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

      [Start] 43.5	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 43.5	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot -0.25} \]
      *-commutative [<=] 43.5	\[ \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot -0.25 \]
      associate-*l* [=>] 43.5	\[ \color{blue}{\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right)} \]
      associate-*r* [=>] 43.5	\[ \color{blue}{\left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)} \cdot \left(\sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right) \]
      *-commutative [=>] 43.5	\[ \color{blue}{\left(\sqrt{8} \cdot \left(\sqrt{2} \cdot y-scale\right)\right)} \cdot \left(\sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot -0.25\right) \]

   if -1.15e27 < y-scale < 8.2000000000000003e-49

   1. Initial program 63.5

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

   2. Taylor expanded in y-scale around 0 54.4

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

   3. Simplified 54.4

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

      [Start] 54.4	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 54.4	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 54.4	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      +-commutative [=>] 54.4	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 54.4	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 54.4	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]

   4. Taylor expanded in b around 0 50.8

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

   5. Simplified 50.8

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

      [Start] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right) \]
      associate-*r* [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\color{blue}{\left(a \cdot \sqrt{2}\right)} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\left(a \cdot \sqrt{2}\right) \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right)\right)\right) \]
      associate-*l* [<=] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\left(a \cdot \sqrt{2}\right) \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}\right)\right)\right) \]
      associate-*l* [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\left(a \cdot \left(\sqrt{2} \cdot \cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)\right)}\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot \cos \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right)\right)\right)\right)\right) \]
      associate-*l* [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot \cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)\right) \]

   6. Taylor expanded in x-scale around 0 50.8

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

   7. Simplified 50.7

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

      [Start] 50.8	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right) \cdot x-scale\right)}\right) \]
      associate-*l* [=>] 50.8	\[ 0.25 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(a \cdot \left(\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right) \cdot x-scale\right)\right)}\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\color{blue}{\left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot x-scale\right)\right)\right) \]
      *-commutative [=>] 50.8	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\left(\sqrt{8} \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right) \cdot x-scale\right)\right)\right) \]
      associate-*l* [=>] 50.7	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\left(\sqrt{8} \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right) \cdot x-scale\right)\right)\right) \]

   8. Applied egg-rr 49.6

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

   9. Simplified 49.6

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

      [Start] 49.6	\[ 0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)}^{2}} \]
      *-commutative [<=] 49.6	\[ 0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{\left(x-scale \cdot \sqrt{8}\right)}\right)\right)}^{2}} \]
      *-commutative [=>] 49.6	\[ 0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2}} \]

   if 8.2000000000000003e-49 < y-scale

   1. Initial program 63.0

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

   2. Taylor expanded in y-scale around -inf 60.0

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

   3. Simplified 59.9

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

      [Start] 60.0	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      associate-*r* [=>] 60.0	\[ \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}} \]
      associate-*r* [=>] 60.0	\[ \left(-0.25 \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \]
      *-commutative [<=] 60.0	\[ \left(-0.25 \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \]
      distribute-lft-out [=>] 60.0	\[ \left(-0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}} \]
      +-commutative [=>] 60.0	\[ \left(-0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}} \]

   4. Taylor expanded in x-scale around -inf 45.8

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

   5. Simplified 45.7

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

      [Start] 45.8	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      associate-*r* [=>] 45.7	\[ \color{blue}{\left(0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} \]
      *-commutative [=>] 45.7	\[ \color{blue}{\sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)} \]

   6. Applied egg-rr 43.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}\right)}^{3}} \cdot \left(\left(\sqrt{8} \cdot \left(\sqrt{2} \cdot y-scale\right)\right) \cdot 0.25\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -1.15 \cdot 10^{+27}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot a, {\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}, {\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 8.2 \cdot 10^{-49}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}}}\right)}^{3} \cdot \left(\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	47.0
Cost	72136

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\\ t_3 := \cos t_0\\ \mathbf{if}\;y-scale \leq -1.15 \cdot 10^{+27}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot a, {t_1}^{2}, {t_3}^{2} \cdot \left(b \cdot b\right)\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{-47}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot 0.25\right) \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(t_3 \cdot b\right)}^{2} + {\left(a \cdot t_1\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 2
Error	47.1
Cost	65864

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\\ t_3 := \cos t_0\\ \mathbf{if}\;y-scale \leq -1.95 \cdot 10^{+27}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot a, {t_1}^{2}, {t_3}^{2} \cdot \left(b \cdot b\right)\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot 0.25\right) \cdot e^{\log \left({\left(t_3 \cdot b\right)}^{2} + {\left(a \cdot t_1\right)}^{2}\right) \cdot 0.5}\\ \end{array} \]

Alternative 3
Error	49.2
Cost	59596

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;x-scale \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.35 \cdot 10^{-256}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.25 \cdot 10^{-97}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot e^{0.5 \cdot \log \left(2 \cdot \left({\left(a \cdot \cos t_0\right)}^{2} + {\left(b \cdot \sin t_0\right)}^{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternative 4
Error	49.1
Cost	59596

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;x-scale \leq -8.5 \cdot 10^{-85}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.5 \cdot 10^{-126}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot {\left({\left(2 \cdot \left({\left(a \cdot \cos t_0\right)}^{2} + {\left(b \cdot \sin t_0\right)}^{2}\right)\right)}^{0.25}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 5
Error	51.1
Cost	33488

\[\begin{array}{l} t_0 := \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;b \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-111}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\right)\right)\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-299}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-47}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot {\left(a \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot t_0\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	50.7
Cost	27024

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.2 \cdot 10^{+27}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale \leq 9 \cdot 10^{-153}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.9 \cdot 10^{-25}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \end{array} \]

Alternative 7
Error	50.8
Cost	26760

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{if}\;y-scale \leq -1.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 8.5 \cdot 10^{-157}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.12 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \end{array} \]

Alternative 8
Error	50.7
Cost	26760

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{if}\;y-scale \leq -1.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-158}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.12 \cdot 10^{-38}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \end{array} \]

Alternative 9
Error	50.7
Cost	26760

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{if}\;y-scale \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 3.1 \cdot 10^{-154}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 10^{-39}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \end{array} \]

Alternative 10
Error	50.3
Cost	14164

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ t_1 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -1.2 \cdot 10^{-94}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.8 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.8 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\right)\right)\\ \end{array} \]

Alternative 11
Error	51.2
Cost	14036

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ t_1 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ t_2 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -8.8 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -6.2 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 1.35 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	51.2
Cost	14036

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ t_1 := 0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ t_2 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -1.9 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.6 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -1.05 \cdot 10^{-289}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 8 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	51.2
Cost	14036

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ t_1 := 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ t_2 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -1.5 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -4.2 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -1.6 \cdot 10^{-289}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 1.5 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 3.3 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	51.2
Cost	14036

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ t_1 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -1.06 \cdot 10^{-94}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.3 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -4.4 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]

Alternative 15
Error	51.3
Cost	14032

\[\begin{array}{l} t_0 := 0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{if}\;x-scale \leq -3.9 \cdot 10^{+115}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{elif}\;x-scale \leq -6 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 9 \cdot 10^{-257}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\right)\right)\\ \end{array} \]

Alternative 16
Error	53.0
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -5.3 \cdot 10^{+38}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.8 \cdot 10^{-37}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \end{array} \]

Alternative 17
Error	54.1
Cost	845

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -5.8 \cdot 10^{+38} \lor \neg \left(y-scale \leq 10^{-37}\right) \land y-scale \leq 5.1 \cdot 10^{+32}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\\ \end{array} \]

Alternative 18
Error	53.7
Cost	448

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

Reproduce

herbie shell --seed 2023056 
(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))))