a from scale-rotated-ellipse

Average Error: 63.4 → 46.6

Time: 2.4min

Precision: binary64

Cost: 65732

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;x-scale \leq -2.7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos t_0}^{2}, b \cdot \left(b \cdot {\sin t_0}^{2}\right)\right)}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 2.22 \cdot 10^{-131}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 2.8 \cdot 10^{-81}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos t_1}^{2}, {\left(b \cdot \sin t_1\right)}^{2}\right)}\right)\right)\\ \end{array} \]

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 (* 0.005555555555555556 angle)))
        (t_1 (* angle (* PI 0.005555555555555556))))
   (if (<= x-scale -2.7e-179)
     (*
      (sqrt 2.0)
      (*
       (*
        (* x-scale (sqrt 8.0))
        (sqrt
         (fma (* a a) (pow (cos t_0) 2.0) (* b (* b (pow (sin t_0) 2.0))))))
       -0.25))
     (if (<= x-scale 2.22e-131)
       (* (* 0.25 y-scale) (* b -4.0))
       (if (<= x-scale 2.8e-81)
         (* (* 0.25 y-scale) (* b 4.0))
         (*
          0.25
          (*
           x-scale
           (*
            (sqrt 8.0)
            (sqrt
             (*
              2.0
              (fma
               (* a a)
               (pow (cos t_1) 2.0)
               (pow (* b (sin 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) * (0.005555555555555556 * angle);
	double t_1 = angle * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (x_45_scale <= -2.7e-179) {
		tmp = sqrt(2.0) * (((x_45_scale * sqrt(8.0)) * sqrt(fma((a * a), pow(cos(t_0), 2.0), (b * (b * pow(sin(t_0), 2.0)))))) * -0.25);
	} else if (x_45_scale <= 2.22e-131) {
		tmp = (0.25 * y_45_scale) * (b * -4.0);
	} else if (x_45_scale <= 2.8e-81) {
		tmp = (0.25 * y_45_scale) * (b * 4.0);
	} else {
		tmp = 0.25 * (x_45_scale * (sqrt(8.0) * sqrt((2.0 * fma((a * a), pow(cos(t_1), 2.0), pow((b * sin(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(0.005555555555555556 * angle))
	t_1 = Float64(angle * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if (x_45_scale <= -2.7e-179)
		tmp = Float64(sqrt(2.0) * Float64(Float64(Float64(x_45_scale * sqrt(8.0)) * sqrt(fma(Float64(a * a), (cos(t_0) ^ 2.0), Float64(b * Float64(b * (sin(t_0) ^ 2.0)))))) * -0.25));
	elseif (x_45_scale <= 2.22e-131)
		tmp = Float64(Float64(0.25 * y_45_scale) * Float64(b * -4.0));
	elseif (x_45_scale <= 2.8e-81)
		tmp = Float64(Float64(0.25 * y_45_scale) * Float64(b * 4.0));
	else
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(sqrt(8.0) * sqrt(Float64(2.0 * fma(Float64(a * a), (cos(t_1) ^ 2.0), (Float64(b * sin(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[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2.7e-179], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] + N[(b * N[(b * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.22e-131], N[(N[(0.25 * y$45$scale), $MachinePrecision] * N[(b * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.8e-81], N[(N[(0.25 * y$45$scale), $MachinePrecision] * N[(b * 4.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x-scale < -2.69999999999999988e-179

   1. Initial program 63.2

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

   2. Simplified 62.8

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot \left(\left(a \cdot \left(-\frac{-8 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{b \cdot a}}\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{\frac{y-scale}{2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}\right)\right)\right)\right)}}{a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot -4\right)\right)} \cdot \left(-{\left(x-scale \cdot y-scale\right)}^{2}\right)} \]
      Proof

      [Start] 63.2

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

   3. Taylor expanded in x-scale around inf 60.3

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

   4. Simplified 60.3

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

      [Start] 60.3	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 60.3	\[ \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}} \]
      associate-*r* [=>] 60.3	\[ \left(0.25 \cdot \color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      distribute-lft-out [=>] 60.3	\[ \left(0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}} \]

   5. Taylor expanded in y-scale around -inf 48.5

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

   6. Simplified 47.2

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

      [Start] 48.5	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 48.5	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot -0.25} \]
      associate-*l* [=>] 48.5	\[ \color{blue}{\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)} \cdot -0.25 \]
      associate-*l* [=>] 48.5	\[ \color{blue}{\sqrt{2} \cdot \left(\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \cdot -0.25\right)} \]

   if -2.69999999999999988e-179 < x-scale < 2.2200000000000001e-131

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0

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

   3. Taylor expanded in angle around 0 47.7

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

   4. Simplified 47.7

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

      [Start] 47.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 47.7	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 47.7	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 47.7	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 52.2

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

   6. Simplified 52.2

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\sqrt{\left(b \cdot b\right) \cdot 16}} \]
      Proof

      [Start] 52.2	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)} \]
      associate-*r* [=>] 52.2	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(2 \cdot 8\right) \cdot \left(b \cdot b\right)}} \]
      *-commutative [=>] 52.2	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(b \cdot b\right) \cdot \left(2 \cdot 8\right)}} \]
      metadata-eval [=>] 52.2	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\left(b \cdot b\right) \cdot \color{blue}{16}} \]

   7. Taylor expanded in b around -inf 47.8

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(-4 \cdot b\right)} \]

   8. Simplified 47.8

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

      [Start] 47.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(-4 \cdot b\right) \]
      *-commutative [=>] 47.8	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(b \cdot -4\right)} \]

   if 2.2200000000000001e-131 < x-scale < 2.7999999999999999e-81

   1. Initial program 63.1

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

   2. Simplified 63.1

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

      [Start] 63.1

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

   3. Taylor expanded in angle around 0 50.3

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

   4. Simplified 50.3

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

      [Start] 50.3	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 50.3	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 50.3	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 50.3	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 55.3

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

   6. Simplified 55.3

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\sqrt{\left(b \cdot b\right) \cdot 16}} \]
      Proof

      [Start] 55.3	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)} \]
      associate-*r* [=>] 55.3	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(2 \cdot 8\right) \cdot \left(b \cdot b\right)}} \]
      *-commutative [=>] 55.3	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(b \cdot b\right) \cdot \left(2 \cdot 8\right)}} \]
      metadata-eval [=>] 55.3	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\left(b \cdot b\right) \cdot \color{blue}{16}} \]

   7. Applied egg-rr 50.2

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(b \cdot 4\right)} \]

   if 2.7999999999999999e-81 < x-scale

   1. Initial program 63.3

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

   2. Simplified 62.8

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot \left(\left(a \cdot \left(-\frac{-8 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{b \cdot a}}\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{\frac{y-scale}{2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}\right)\right)\right)\right)}}{a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot -4\right)\right)} \cdot \left(-{\left(x-scale \cdot y-scale\right)}^{2}\right)} \]
      Proof

      [Start] 63.3

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

   3. Taylor expanded in y-scale around 0 46.9

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

   4. Simplified 46.9

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

      [Start] 46.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 46.9	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{a \cdot a}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]
      *-commutative [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right)\right) \]

   5. Applied egg-rr 54.4

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)} - 1\right)}\right)\right) \]

   6. Simplified 44.3

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

      [Start] 54.4	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)} - 1\right)\right)\right) \]
      expm1-def [=>] 47.3	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right)}\right)\right) \]
      expm1-log1p [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}}\right)\right) \]
      *-commutative [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right) \]
      associate-*l* [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right) \]
      *-commutative [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}, \color{blue}{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 46.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}, \left(b \cdot b\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}\right)\right) \]
      swap-sqr [<=] 44.3	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}, \color{blue}{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}\right)\right) \]
      unpow2 [<=] 44.3	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}, \color{blue}{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 46.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -2.7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}, b \cdot \left(b \cdot {\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}\right)\right)}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 2.22 \cdot 10^{-131}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 2.8 \cdot 10^{-81}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}, {\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	48.3
Cost	59732

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_2 := a \cdot \left(\sqrt{8} \cdot \cos t_1\right)\\ t_3 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ t_4 := \cos t_0\\ t_5 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {t_4}^{2}, {\left(b \cdot \sin t_0\right)}^{2}\right)}\right)\right)\\ \mathbf{if}\;y-scale \leq -4.8 \cdot 10^{+44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq -3.2 \cdot 10^{-21}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y-scale \leq -2.85 \cdot 10^{-67}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq -6.9 \cdot 10^{-227}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot \left(a \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -7.1 \cdot 10^{-262}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y-scale \leq 9 \cdot 10^{-292}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot t_2\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{-108}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.9 \cdot 10^{-82}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(t_4 \cdot \frac{\sqrt{2}}{y-scale}\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.6 \cdot 10^{-61}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 2
Error	48.4
Cost	59732

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0\\ t_2 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ t_3 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_4 := \cos t_3\\ t_5 := a \cdot \left(\sqrt{8} \cdot t_4\right)\\ \mathbf{if}\;y-scale \leq -2.4 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq -3.4 \cdot 10^{-21}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {t_4}^{2}, {\sin t_3}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq -8.2 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq -9.6 \cdot 10^{-225}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(\sqrt{8} \cdot \left(a \cdot \cos \left({\left(\sqrt[3]{t_3}\right)}^{3}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -4.4 \cdot 10^{-258}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {t_1}^{2}, {\left(b \cdot \sin t_0\right)}^{2}\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.6 \cdot 10^{-291}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot t_5\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.2 \cdot 10^{-80}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(t_1 \cdot \frac{\sqrt{2}}{y-scale}\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.6 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 3
Error	48.2
Cost	40140

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;x-scale \leq -5.6 \cdot 10^{-36}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8 \cdot {\left(a \cdot \cos t_0\right)}^{2}}\right)\\ \mathbf{elif}\;x-scale \leq 3.1 \cdot 10^{-131}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, 1, {\sin t_0}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right)\\ \end{array} \]

Alternative 4
Error	51.6
Cost	27024

\[\begin{array}{l} t_0 := -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ t_1 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{if}\;b \leq -2.95 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.62 \cdot 10^{-243}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	51.5
Cost	27024

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ t_1 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-243}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	51.6
Cost	27024

\[\begin{array}{l} t_0 := x-scale \cdot \sqrt{2}\\ t_1 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ t_2 := \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;b \leq -7.6 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-248}:\\ \;\;\;\;-0.25 \cdot \left(t_0 \cdot \left(\sqrt{8} \cdot \left(a \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-243}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+36}:\\ \;\;\;\;-0.25 \cdot \left(t_0 \cdot \left(t_2 \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	51.6
Cost	27024

\[\begin{array}{l} t_0 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{2} \cdot \left(-0.25 \cdot \left(\left(\sqrt{8} \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(x-scale \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-242}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+37}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	51.6
Cost	13904

\[\begin{array}{l} t_0 := \left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ t_1 := \sqrt{2} \cdot a\\ t_2 := -0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot t_1\right)\\ \mathbf{if}\;b \leq -3 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-245}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot t_1\right)\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	51.6
Cost	13641

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-25} \lor \neg \left(b \leq 1.25 \cdot 10^{+37}\right):\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	51.6
Cost	13641

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-23} \lor \neg \left(b \leq 1.35 \cdot 10^{+37}\right):\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	51.7
Cost	13641

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-26} \lor \neg \left(b \leq 2.4 \cdot 10^{+37}\right):\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]

Alternative 12
Error	53.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 3.8 \cdot 10^{-132}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot y-scale\right) \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 13
Error	53.8
Cost	448

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

Reproduce

herbie shell --seed 2023187 
(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))))