a from scale-rotated-ellipse

Average Error: 99.24% → 77.04%

Time: 1.8min

Precision: binary64

Cost: 85708

Math

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

↓

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := {\sin t_1}^{2}\\ t_3 := \cos t_1\\ t_4 := {t_3}^{2}\\ \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{-89}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos t_0}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin t_0}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq -2.9 \cdot 10^{-213}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{\frac{y-scale}{t_3}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, t_4, \left(b \cdot b\right) \cdot t_2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, t_4, \mathsf{fma}\left(2, \left(a \cdot a\right) \cdot t_2, \left(b \cdot b\right) \cdot t_4\right)\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (+
      (+
       (/
        (/
         (+
          (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
         x-scale)
        x-scale)
       (/
        (/
         (+
          (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
         y-scale)
        y-scale))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
            x-scale)
           x-scale)
          (/
           (/
            (+
             (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
            y-scale)
           y-scale))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 2.0 (- (pow b 2.0) (pow a 2.0)))
             (sin (* (/ angle 180.0) PI)))
            (cos (* (/ angle 180.0) PI)))
           x-scale)
          y-scale)
         2.0)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (pow (sin t_1) 2.0))
        (t_3 (cos t_1))
        (t_4 (pow t_3 2.0)))
   (if (<= y-scale -6.5e-89)
     (*
      (* -0.25 (* (* y-scale (sqrt 2.0)) (sqrt 8.0)))
      (sqrt (fma (pow (cos t_0) 2.0) (* b b) (* (* a a) (pow (sin t_0) 2.0)))))
     (if (<= y-scale -2.9e-213)
       (*
        -0.25
        (*
         y-scale
         (* (* (sqrt 8.0) (* a x-scale)) (/ (sqrt 2.0) (/ y-scale t_3)))))
       (if (<= y-scale 1.75e-84)
         (*
          0.25
          (*
           x-scale
           (* (sqrt 8.0) (sqrt (* 2.0 (fma (* a a) t_4 (* (* b b) t_2)))))))
         (*
          0.25
          (*
           y-scale
           (*
            (sqrt 8.0)
            (sqrt
             (fma
              (* b b)
              t_4
              (fma 2.0 (* (* a a) t_2) (* (* b b) t_4))))))))))))

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 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = pow(sin(t_1), 2.0);
	double t_3 = cos(t_1);
	double t_4 = pow(t_3, 2.0);
	double tmp;
	if (y_45_scale <= -6.5e-89) {
		tmp = (-0.25 * ((y_45_scale * sqrt(2.0)) * sqrt(8.0))) * sqrt(fma(pow(cos(t_0), 2.0), (b * b), ((a * a) * pow(sin(t_0), 2.0))));
	} else if (y_45_scale <= -2.9e-213) {
		tmp = -0.25 * (y_45_scale * ((sqrt(8.0) * (a * x_45_scale)) * (sqrt(2.0) / (y_45_scale / t_3))));
	} else if (y_45_scale <= 1.75e-84) {
		tmp = 0.25 * (x_45_scale * (sqrt(8.0) * sqrt((2.0 * fma((a * a), t_4, ((b * b) * t_2))))));
	} else {
		tmp = 0.25 * (y_45_scale * (sqrt(8.0) * sqrt(fma((b * b), t_4, fma(2.0, ((a * a) * t_2), ((b * b) * t_4))))));
	}
	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(Float64(0.005555555555555556 * angle) * pi)
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1) ^ 2.0
	t_3 = cos(t_1)
	t_4 = t_3 ^ 2.0
	tmp = 0.0
	if (y_45_scale <= -6.5e-89)
		tmp = Float64(Float64(-0.25 * Float64(Float64(y_45_scale * sqrt(2.0)) * sqrt(8.0))) * sqrt(fma((cos(t_0) ^ 2.0), Float64(b * b), Float64(Float64(a * a) * (sin(t_0) ^ 2.0)))));
	elseif (y_45_scale <= -2.9e-213)
		tmp = Float64(-0.25 * Float64(y_45_scale * Float64(Float64(sqrt(8.0) * Float64(a * x_45_scale)) * Float64(sqrt(2.0) / Float64(y_45_scale / t_3)))));
	elseif (y_45_scale <= 1.75e-84)
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(sqrt(8.0) * sqrt(Float64(2.0 * fma(Float64(a * a), t_4, Float64(Float64(b * b) * t_2)))))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale * Float64(sqrt(8.0) * sqrt(fma(Float64(b * b), t_4, fma(2.0, Float64(Float64(a * a) * t_2), Float64(Float64(b * b) * t_4)))))));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, If[LessEqual[y$45$scale, -6.5e-89], N[(N[(-0.25 * N[(N[(y$45$scale * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -2.9e-213], N[(-0.25 * N[(y$45$scale * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(a * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(y$45$scale / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 1.75e-84], N[(0.25 * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] * t$95$4 + N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[N[(N[(b * b), $MachinePrecision] * t$95$4 + N[(2.0 * N[(N[(a * a), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < -6.50000000000000034e-89

   1. Initial program 98.92

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

   2. Simplified 98.06

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

      [Start] 98.92

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

   3. Taylor expanded in y-scale around -inf 96.81

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

   4. Simplified 96.82

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

      [Start] 96.81	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right)\right) \]
      *-commutative [=>] 96.81	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}\right)}\right) \]
      *-commutative [=>] 96.81	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   5. Taylor expanded in x-scale around 0 73.14

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

   6. Simplified 73.1

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

      [Start] 73.14	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      associate-*r* [=>] 73.09	\[ \color{blue}{\left(-0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} \]
      associate-*r* [=>] 73.09	\[ \left(-0.25 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \]
      fma-def [=>] 73.09	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]
      *-commutative [=>] 73.09	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      associate-*r* [<=] 73.09	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      *-commutative [<=] 73.09	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      associate-*r* [=>] 73.1	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      *-commutative [<=] 73.1	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      unpow2 [=>] 73.1	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}, \color{blue}{b \cdot b}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]

   if -6.50000000000000034e-89 < y-scale < -2.8999999999999999e-213

   1. Initial program 99.76

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

   2. Simplified 99.56

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

      [Start] 99.76

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

   3. Taylor expanded in a around -inf 100

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

   4. Simplified 100

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

      [Start] 100

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

      associate-*l* [=>] 100

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

   5. Taylor expanded in x-scale around inf 83.91

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

   6. Simplified 83.91

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

      [Start] 83.91	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \frac{\sqrt{2} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale}\right)\right) \]
      associate-/l* [=>] 83.91	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \color{blue}{\frac{\sqrt{2}}{\frac{y-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}}\right)\right) \]

   if -2.8999999999999999e-213 < y-scale < 1.7500000000000001e-84

   1. Initial program 99.75

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

   2. Simplified 99.58

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

      [Start] 99.75

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

   3. Taylor expanded in y-scale around 0 82.41

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

   4. Simplified 82.39

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

      [Start] 82.41	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 82.39	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 82.39	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 82.39	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 82.39	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{a \cdot a}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]
      *-commutative [=>] 82.39	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 82.39	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right)\right) \]

   if 1.7500000000000001e-84 < y-scale

   1. Initial program 98.83

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

   2. Simplified 98.4

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

      [Start] 98.83

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

   3. Taylor expanded in x-scale around 0 72.81

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

   4. Simplified 72.77

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

      [Start] 72.81	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      associate-*l* [=>] 72.77	\[ 0.25 \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
      fma-def [=>] 72.77	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{\mathsf{fma}\left({b}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}}\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 77.04

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{-89}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq -2.9 \cdot 10^{-213}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{\frac{y-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \mathsf{fma}\left(2, \left(a \cdot a\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	77.05%
Cost	72396

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := {\sin t_1}^{2}\\ t_3 := \cos t_1\\ t_4 := {t_3}^{2}\\ \mathbf{if}\;y-scale \leq -2.22 \cdot 10^{-85}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos t_0}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin t_0}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq -2.4 \cdot 10^{-213}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{\frac{y-scale}{t_3}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.8 \cdot 10^{-84}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, t_4, \left(b \cdot b\right) \cdot t_2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{t_4 \cdot {b}^{2} + t_2 \cdot {a}^{2}}\right)\\ \end{array} \]

Alternative 2
Error	78.18%
Cost	65732

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t_1\\ \mathbf{if}\;y-scale \leq -2 \cdot 10^{-86}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos t_0}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin t_0}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq -2.65 \cdot 10^{-213}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{\frac{y-scale}{t_2}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.9 \cdot 10^{-84}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {t_2}^{2}, \left(b \cdot b\right) \cdot {\sin t_1}^{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 3
Error	80.07%
Cost	59860

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\left(y-scale \cdot x-scale\right)}^{2}\\ t_2 := y-scale \cdot \left(x-scale \cdot x-scale\right)\\ \mathbf{if}\;x-scale \leq -1.18 \cdot 10^{-169}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\left(y-scale \cdot b\right) \cdot \left(\sqrt{8} \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.65 \cdot 10^{-216}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{elif}\;x-scale \leq 1.4 \cdot 10^{-80}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;t_1 \cdot \left(0.25 \cdot \left|\frac{b \cdot 4}{t_2}\right|\right)\\ \mathbf{elif}\;x-scale \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;2 - b \cdot \frac{t_1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos t_0}^{2}, \left(b \cdot b\right) \cdot {\sin t_0}^{2}\right)}\right)\right)\\ \end{array} \]

Alternative 4
Error	84.97%
Cost	14028

\[\begin{array}{l} t_0 := y-scale \cdot \left(-b\right)\\ \mathbf{if}\;angle \leq -2.5 \cdot 10^{-199}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;angle \leq -2.6 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq -1.05 \cdot 10^{-277}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{elif}\;angle \leq 1.75 \cdot 10^{-275} \lor \neg \left(angle \leq 6.8 \cdot 10^{-146}\right) \land angle \leq 4.6 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 5
Error	84.15%
Cost	976

\[\begin{array}{l} t_0 := y-scale \cdot \left(-b\right)\\ \mathbf{if}\;x-scale \leq -2.4 \cdot 10^{-171}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 7.5 \cdot 10^{-70}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 1.65 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y-scale \cdot b + -1\right)\\ \end{array} \]

Alternative 6
Error	84.07%
Cost	785

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 9.6 \cdot 10^{-216} \lor \neg \left(x-scale \leq 4.9 \cdot 10^{-70}\right) \land x-scale \leq 8.5 \cdot 10^{+80}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 7
Error	84.01%
Cost	192

\[y-scale \cdot b \]

Reproduce

herbie shell --seed 2023121 
(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))))