a from scale-rotated-ellipse

Average Error: 63.5 → 46.4

Time: 2.3min

Precision: binary64

Cost: 91920

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := b \cdot \cos t_0\\ t_2 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_3 := y-scale \cdot \sqrt{8}\\ \mathbf{if}\;y-scale \leq -5.8 \cdot 10^{-67}:\\ \;\;\;\;-0.25 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos t_2}^{2}, \left(a \cdot a\right) \cdot {\sin t_2}^{2}\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.45 \cdot 10^{-263}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-190}:\\ \;\;\;\;\left(\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right) \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.55 \cdot 10^{+41}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_3 \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(t_1, \sqrt{\mathsf{fma}\left(2, {\left(a \cdot \sin t_0\right)}^{2}, {t_1}^{2}\right)}\right)}\right)}^{3}\right)\\ \end{array} \]

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 (* b (cos t_0)))
        (t_2 (* angle (* 0.005555555555555556 PI)))
        (t_3 (* y-scale (sqrt 8.0))))
   (if (<= y-scale -5.8e-67)
     (*
      -0.25
      (*
       t_3
       (*
        (sqrt 2.0)
        (sqrt
         (fma (* b b) (pow (cos t_2) 2.0) (* (* a a) (pow (sin t_2) 2.0)))))))
     (if (<= y-scale 1.45e-263)
       (* 0.25 (* (sqrt 2.0) (* x-scale (* (sqrt 8.0) a))))
       (if (<= y-scale 7e-190)
         (*
          (* (* -0.25 x-scale) (* (sqrt 2.0) a))
          (* (sqrt 8.0) (cos (* 0.005555555555555556 (* angle PI)))))
         (if (<= y-scale 2.55e+41)
           (* (* y-scale 0.25) (* (sqrt 2.0) (fabs (* (sqrt 8.0) b))))
           (*
            0.25
            (*
             t_3
             (pow
              (cbrt
               (hypot
                t_1
                (sqrt (fma 2.0 (pow (* a (sin t_0)) 2.0) (pow t_1 2.0)))))
              3.0)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) + (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)) + sqrt((pow(((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) - (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = b * cos(t_0);
	double t_2 = angle * (0.005555555555555556 * ((double) M_PI));
	double t_3 = y_45_scale * sqrt(8.0);
	double tmp;
	if (y_45_scale <= -5.8e-67) {
		tmp = -0.25 * (t_3 * (sqrt(2.0) * sqrt(fma((b * b), pow(cos(t_2), 2.0), ((a * a) * pow(sin(t_2), 2.0))))));
	} else if (y_45_scale <= 1.45e-263) {
		tmp = 0.25 * (sqrt(2.0) * (x_45_scale * (sqrt(8.0) * a)));
	} else if (y_45_scale <= 7e-190) {
		tmp = ((-0.25 * x_45_scale) * (sqrt(2.0) * a)) * (sqrt(8.0) * cos((0.005555555555555556 * (angle * ((double) M_PI)))));
	} else if (y_45_scale <= 2.55e+41) {
		tmp = (y_45_scale * 0.25) * (sqrt(2.0) * fabs((sqrt(8.0) * b)));
	} else {
		tmp = 0.25 * (t_3 * pow(cbrt(hypot(t_1, sqrt(fma(2.0, pow((a * sin(t_0)), 2.0), pow(t_1, 2.0))))), 3.0));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) + sqrt(Float64((Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0))))))) / Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64(b * cos(t_0))
	t_2 = Float64(angle * Float64(0.005555555555555556 * pi))
	t_3 = Float64(y_45_scale * sqrt(8.0))
	tmp = 0.0
	if (y_45_scale <= -5.8e-67)
		tmp = Float64(-0.25 * Float64(t_3 * Float64(sqrt(2.0) * sqrt(fma(Float64(b * b), (cos(t_2) ^ 2.0), Float64(Float64(a * a) * (sin(t_2) ^ 2.0)))))));
	elseif (y_45_scale <= 1.45e-263)
		tmp = Float64(0.25 * Float64(sqrt(2.0) * Float64(x_45_scale * Float64(sqrt(8.0) * a))));
	elseif (y_45_scale <= 7e-190)
		tmp = Float64(Float64(Float64(-0.25 * x_45_scale) * Float64(sqrt(2.0) * a)) * Float64(sqrt(8.0) * cos(Float64(0.005555555555555556 * Float64(angle * pi)))));
	elseif (y_45_scale <= 2.55e+41)
		tmp = Float64(Float64(y_45_scale * 0.25) * Float64(sqrt(2.0) * abs(Float64(sqrt(8.0) * b))));
	else
		tmp = Float64(0.25 * Float64(t_3 * (cbrt(hypot(t_1, sqrt(fma(2.0, (Float64(a * sin(t_0)) ^ 2.0), (t_1 ^ 2.0))))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[((-N[Sqrt[N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] - N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -5.8e-67], N[(-0.25 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(b * b), $MachinePrecision] * N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 1.45e-263], N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 7e-190], N[(N[(N[(-0.25 * x$45$scale), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 2.55e+41], N[(N[(y$45$scale * 0.25), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[N[(N[Sqrt[8.0], $MachinePrecision] * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$3 * N[Power[N[Power[N[Sqrt[t$95$1 ^ 2 + N[Sqrt[N[(2.0 * N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < -5.8000000000000001e-67

   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 62.7

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

      [Start] 63.1

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

   3. Taylor expanded in y-scale around -inf 59.9

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

   4. Simplified 59.9

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

      [Start] 59.9	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      associate-*r* [=>] 59.9	\[ -0.25 \cdot \left(\color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)} \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      *-commutative [=>] 59.9	\[ -0.25 \cdot \left(\left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      distribute-lft-out [=>] 59.9	\[ -0.25 \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}}\right) \]

   5. Taylor expanded in x-scale around 0 46.5

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

   6. Simplified 46.5

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

      [Start] 46.5	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      *-commutative [=>] 46.5	\[ -0.25 \cdot \left(\color{blue}{\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      associate-*l* [=>] 46.5	\[ -0.25 \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)} \]
      *-commutative [<=] 46.5	\[ -0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right) \]

   if -5.8000000000000001e-67 < y-scale < 1.45000000000000002e-263

   1. Initial program 63.7

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

   2. Simplified 63.7

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

      [Start] 63.7

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

   3. Taylor expanded in x-scale around inf 61.8

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

   4. Simplified 61.7

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

      [Start] 61.8	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      associate-*r* [=>] 61.8	\[ \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}} \]
      associate-*r* [=>] 61.8	\[ \left(0.25 \cdot \color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \]
      distribute-lft-out [=>] 61.8	\[ \left(0.25 \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}} \]

   5. Taylor expanded in angle around 0 49.3

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

   if 1.45000000000000002e-263 < y-scale < 6.9999999999999999e-190

   1. Initial program 63.9

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

   2. Simplified 64.0

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

      [Start] 63.9

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

   3. Taylor expanded in a around -inf 64.0

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\frac{a \cdot \sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)}\right) \]

   4. Simplified 64.0

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{\frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}}, {\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}\right)} \cdot \left(\frac{a}{x-scale} \cdot \frac{\sqrt{8}}{y-scale}\right)\right)}\right) \]
      Proof

      [Start] 64.0	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\frac{a \cdot \sqrt{8}}{y-scale \cdot x-scale} \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)\right) \]
      *-commutative [=>] 64.0	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)} \cdot \frac{a \cdot \sqrt{8}}{y-scale \cdot x-scale}\right)}\right) \]
      *-commutative [<=] 64.0	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)} \cdot \frac{a \cdot \sqrt{8}}{\color{blue}{x-scale \cdot y-scale}}\right)\right) \]

   5. Taylor expanded in x-scale around inf 50.3

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

   6. Simplified 50.3

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

      [Start] 50.3	\[ -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      associate-*r* [=>] 50.3	\[ \color{blue}{\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)} \]
      associate-*r* [=>] 50.3	\[ \left(-0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)} \]
      associate-*r* [=>] 50.3	\[ \color{blue}{\left(\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)} \]
      *-commutative [=>] 50.3	\[ \left(\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right) \cdot \color{blue}{\left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 6.9999999999999999e-190 < y-scale < 2.54999999999999989e41

   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. Simplified 63.5

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

      [Start] 63.4

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

   3. Taylor expanded in angle around 0 54.5

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

   4. Simplified 54.5

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

      [Start] 54.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 54.5	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 54.5	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 54.5	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 53.8

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

   6. Simplified 47.5

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

      [Start] 53.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{8 \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      rem-square-sqrt [<=] 53.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot \sqrt{8}\right)} \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      swap-sqr [<=] 53.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{8} \cdot b\right)}} \cdot \sqrt{2}\right) \]
      rem-sqrt-square [=>] 47.5	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left|\sqrt{8} \cdot b\right|} \cdot \sqrt{2}\right) \]
      *-commutative [=>] 47.5	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\left|\color{blue}{b \cdot \sqrt{8}}\right| \cdot \sqrt{2}\right) \]

   if 2.54999999999999989e41 < y-scale

   1. Initial program 63.6

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

   2. Simplified 63.2

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

      [Start] 63.6

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

   3. Taylor expanded in x-scale around 0 42.0

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

   4. Simplified 42.0

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

      [Start] 42.0	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      fma-def [=>] 42.0	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left({b}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}}\right) \]
      unpow2 [=>] 42.0	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{b \cdot b}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      *-commutative [=>] 42.0	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + \color{blue}{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)}\right) \]
      fma-def [=>] 42.0	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

   5. Applied egg-rr 38.6

      \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b, \sqrt{\mathsf{fma}\left(2, {\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}, {\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{3}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 46.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -5.8 \cdot 10^{-67}:\\ \;\;\;\;-0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}, \left(a \cdot a\right) \cdot {\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.45 \cdot 10^{-263}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-190}:\\ \;\;\;\;\left(\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right) \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.55 \cdot 10^{+41}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(b \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \sqrt{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}, {\left(b \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)}\right)}\right)}^{3}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	46.5
Cost	65732

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_2 := y-scale \cdot \sqrt{8}\\ \mathbf{if}\;y-scale \leq -7.2 \cdot 10^{-67}:\\ \;\;\;\;-0.25 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos t_1}^{2}, \left(a \cdot a\right) \cdot {\sin t_1}^{2}\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.1 \cdot 10^{-263}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\left(\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot a\right)\right) \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \sqrt{2 \cdot \left({\left(b \cdot \cos t_0\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 2
Error	48.8
Cost	53068

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;y-scale \leq -2.45 \cdot 10^{-197}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{8} \cdot \left(\left(\sqrt{2} \cdot x-scale\right) \cdot \left(a \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(b \cdot t_1\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 3
Error	48.7
Cost	26760

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -3.1 \cdot 10^{-197}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 5.8 \cdot 10^{-191}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{8} \cdot \left(\left(\sqrt{2} \cdot x-scale\right) \cdot \left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 4
Error	48.7
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -2.35 \cdot 10^{-197}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 8.5 \cdot 10^{-191}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 5
Error	48.7
Cost	19976

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 1.85 \cdot 10^{-189}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot {\left(\left|\sqrt[3]{b \cdot 4}\right|\right)}^{3}\\ \end{array} \]

Alternative 6
Error	51.2
Cost	13904

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -2.45 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.6 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 7
Error	52.0
Cost	13640

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -2 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.35 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 0.052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.8 \cdot 10^{+197}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 8
Error	52.0
Cost	13640

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -3.1 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.2 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 0.14:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{+195}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 9
Error	53.7
Cost	845

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+129} \lor \neg \left(b \leq 3.7 \cdot 10^{+77}\right) \land b \leq 4.8 \cdot 10^{+231}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \end{array} \]

Alternative 10
Error	53.9
Cost	448

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

Reproduce

herbie shell --seed 2023012 
(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))))