a from scale-rotated-ellipse

Average Error: 63.5 → 41.8

Time: 1.9min

Precision: binary64

Cost: 65732

Math

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

↓

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ \mathbf{if}\;y-scale \leq -2.8 \cdot 10^{+105}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos t_0}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin t_0}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq 3.05 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

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)))
   (if (<= y-scale -2.8e+105)
     (*
      (* -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 3.05e-59)
       (* 0.25 (fabs (* a (* (sqrt 2.0) (* (sqrt 8.0) x-scale)))))
       (* (* y-scale 0.25) (* (sqrt 2.0) (fabs (* (sqrt 8.0) b))))))))

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 tmp;
	if (y_45_scale <= -2.8e+105) {
		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 <= 3.05e-59) {
		tmp = 0.25 * fabs((a * (sqrt(2.0) * (sqrt(8.0) * x_45_scale))));
	} else {
		tmp = (y_45_scale * 0.25) * (sqrt(2.0) * fabs((sqrt(8.0) * b)));
	}
	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)
	tmp = 0.0
	if (y_45_scale <= -2.8e+105)
		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 <= 3.05e-59)
		tmp = Float64(0.25 * abs(Float64(a * Float64(sqrt(2.0) * Float64(sqrt(8.0) * x_45_scale)))));
	else
		tmp = Float64(Float64(y_45_scale * 0.25) * Float64(sqrt(2.0) * abs(Float64(sqrt(8.0) * b))));
	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]}, If[LessEqual[y$45$scale, -2.8e+105], 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, 3.05e-59], N[(0.25 * N[Abs[N[(a * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -2.8000000000000001e105

   1. Initial program 63.8

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

   2. Simplified 63.7

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

      [Start] 63.8

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

   3. Taylor expanded in y-scale around -inf 62.5

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

   4. Simplified 62.5

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

      [Start] 62.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right)\right) \]
      *-commutative [=>] 62.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}\right)}\right) \]
      *-commutative [=>] 62.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(\sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   5. Taylor expanded in x-scale around 0 39.9

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

   6. Simplified 39.9

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

      [Start] 39.9	\[ -0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right) \]
      associate-*r* [=>] 39.9	\[ \color{blue}{\left(-0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} \]
      associate-*r* [=>] 39.9	\[ \left(-0.25 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)}\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \]
      fma-def [=>] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]
      *-commutative [=>] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      associate-*r* [<=] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      *-commutative [<=] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      associate-*r* [=>] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      *-commutative [<=] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)}^{2}, {b}^{2}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]
      unpow2 [=>] 39.9	\[ \left(-0.25 \cdot \left(\left(\sqrt{2} \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}, \color{blue}{b \cdot b}, {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \]

   if -2.8000000000000001e105 < y-scale < 3.0499999999999998e-59

   1. Initial program 63.5

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

   2. Simplified 63.1

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

      [Start] 63.5

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

   3. Taylor expanded in a around inf 62.7

      \[\leadsto {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(-1 \cdot \left(\frac{a \cdot \sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)\right)}\right) \]

   4. Simplified 62.7

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

      [Start] 62.7

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-1 \cdot \left(\frac{a \cdot \sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)\right)\right) \]

      associate-*r* [=>] 62.7

      \[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(\left(-1 \cdot \frac{a \cdot \sqrt{8}}{x-scale \cdot y-scale}\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)}\right) \]

   5. Taylor expanded in angle around 0 51.9

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

   6. Simplified 51.9

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

      [Start] 51.9	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 51.9	\[ 0.25 \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \left(a \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 51.9	\[ 0.25 \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot \sqrt{2}\right)\right)} \]
      *-commutative [<=] 51.9	\[ 0.25 \cdot \left(\left(a \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot x-scale\right)}\right) \]

   7. Applied egg-rr 42.1

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

   8. Applied egg-rr 54.8

      \[\leadsto 0.25 \cdot \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right)} + -1}\right| \]

   9. Simplified 42.2

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

      [Start] 54.8	\[ 0.25 \cdot \left|e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right)} + -1\right| \]
      metadata-eval [<=] 54.8	\[ 0.25 \cdot \left|e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right)} + \color{blue}{\left(-1\right)}\right| \]
      sub-neg [<=] 54.8	\[ 0.25 \cdot \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right)} - 1}\right| \]
      expm1-def [=>] 48.8	\[ 0.25 \cdot \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right)\right)}\right| \]
      expm1-log1p [=>] 42.1	\[ 0.25 \cdot \left|\color{blue}{\sqrt{2} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)}\right| \]
      associate-*r* [=>] 42.2	\[ 0.25 \cdot \left|\color{blue}{\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot a}\right| \]

   if 3.0499999999999998e-59 < y-scale

   1. Initial program 63.2

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

   2. Simplified 62.6

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

      [Start] 63.2

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

   3. Taylor expanded in angle around 0 52.8

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

   4. Simplified 52.8

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

      [Start] 52.8	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 52.8	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 52.8	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 52.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 48.7

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

   6. Simplified 41.6

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

      [Start] 48.7	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{8 \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      rem-square-sqrt [<=] 48.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot \sqrt{8}\right)} \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      swap-sqr [<=] 48.8	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{8} \cdot b\right)}} \cdot \sqrt{2}\right) \]
      rem-sqrt-square [=>] 41.6	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left|\sqrt{8} \cdot b\right|} \cdot \sqrt{2}\right) \]
      *-commutative [<=] 41.6	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\left|\color{blue}{b \cdot \sqrt{8}}\right| \cdot \sqrt{2}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 41.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -2.8 \cdot 10^{+105}:\\ \;\;\;\;\left(-0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left({\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}\right)}\\ \mathbf{elif}\;y-scale \leq 3.05 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	41.8
Cost	65732

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ \mathbf{if}\;y-scale \leq -5.2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left({\cos t_0}^{2}, b \cdot b, \left(a \cdot a\right) \cdot {\sin t_0}^{2}\right)} \cdot \left(-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3 \cdot 10^{-59}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 2
Error	43.0
Cost	26628

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -2.9 \cdot 10^{+103}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 3
Error	46.1
Cost	20173

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -3.6 \cdot 10^{+101} \lor \neg \left(y-scale \leq 1.05 \cdot 10^{-11}\right) \land y-scale \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sqrt{2} \cdot x-scale\right) \cdot \left(0.25 \cdot \left(\sqrt{8} \cdot a\right)\right)\right|\\ \end{array} \]

Alternative 4
Error	46.2
Cost	20173

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -2.2 \cdot 10^{+101} \lor \neg \left(y-scale \leq 8.5 \cdot 10^{-12}\right) \land y-scale \leq 5.9 \cdot 10^{+110}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \end{array} \]

Alternative 5
Error	46.2
Cost	20172

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ t_1 := \sqrt{8} \cdot a\\ \mathbf{if}\;y-scale \leq -4 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.45 \cdot 10^{-12}:\\ \;\;\;\;\left|\left(\sqrt{2} \cdot x-scale\right) \cdot \left(0.25 \cdot t_1\right)\right|\\ \mathbf{elif}\;y-scale \leq 9.5 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left|x-scale \cdot \left(\sqrt{2} \cdot t_1\right)\right|\\ \end{array} \]

Alternative 6
Error	46.1
Cost	20172

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ t_1 := \sqrt{8} \cdot a\\ \mathbf{if}\;y-scale \leq -2.6 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.35 \cdot 10^{-11}:\\ \;\;\;\;0.25 \cdot \left|\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right|\\ \mathbf{elif}\;y-scale \leq 1.6 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left|x-scale \cdot \left(\sqrt{2} \cdot t_1\right)\right|\\ \end{array} \]

Alternative 7
Error	42.9
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -4.4 \cdot 10^{+101}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;0.25 \cdot \left|a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot b\right|\right)\\ \end{array} \]

Alternative 8
Error	51.3
Cost	14168

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ t_1 := -0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ t_2 := 0.25 \cdot \left(\left(\sqrt{8} \cdot a\right) \cdot \left(\sqrt{2} \cdot x-scale\right)\right)\\ \mathbf{if}\;y-scale \leq -1.28 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -1950000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq -9 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 9.5 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \mathbf{elif}\;y-scale \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	51.0
Cost	14164

\[\begin{array}{l} t_0 := \sqrt{2} \cdot x-scale\\ t_1 := \sqrt{8} \cdot a\\ \mathbf{if}\;x-scale \leq -1.25 \cdot 10^{+87}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot t_0\right)\\ \mathbf{elif}\;x-scale \leq -3.3 \cdot 10^{-16}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -2.9 \cdot 10^{-140}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right)\\ \mathbf{elif}\;x-scale \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 1.95 \cdot 10^{+59}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \sqrt{a \cdot \left(8 \cdot a\right)}\right)\\ \end{array} \]

Alternative 10
Error	51.0
Cost	14164

\[\begin{array}{l} t_0 := \sqrt{2} \cdot x-scale\\ t_1 := \sqrt{8} \cdot a\\ \mathbf{if}\;x-scale \leq -1.5 \cdot 10^{+90}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot t_0\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-16}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -4.2 \cdot 10^{-140}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_1\right)\right)\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{-157}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 1.85 \cdot 10^{+59}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \sqrt{x-scale \cdot \left(2 \cdot x-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \sqrt{a \cdot \left(8 \cdot a\right)}\right)\\ \end{array} \]

Alternative 11
Error	51.6
Cost	13772

\[\begin{array}{l} t_0 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{if}\;x-scale \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -1.6 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.9 \cdot 10^{-157}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \end{array} \]

Alternative 12
Error	51.6
Cost	13772

\[\begin{array}{l} t_0 := \sqrt{8} \cdot a\\ \mathbf{if}\;x-scale \leq -4.7 \cdot 10^{+86}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -1.05 \cdot 10^{-140}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.3 \cdot 10^{-157}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \end{array} \]

Alternative 13
Error	51.6
Cost	13772

\[\begin{array}{l} t_0 := \sqrt{8} \cdot a\\ \mathbf{if}\;x-scale \leq -1.25 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot x-scale\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.75 \cdot 10^{-16}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -1 \cdot 10^{-139}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq 4.9 \cdot 10^{-157}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \end{array} \]

Alternative 14
Error	50.3
Cost	13320

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -5.1 \cdot 10^{+89}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;y-scale \leq 10^{-207}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 15
Error	52.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.9 \cdot 10^{+60}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;y-scale \leq 1.65 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\left(\left(a \cdot a\right) \cdot 16\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot 0.0625\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 16
Error	52.0
Cost	7368

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.2 \cdot 10^{+60}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;y-scale \leq 1.92 \cdot 10^{-209}:\\ \;\;\;\;0.25 \cdot \sqrt{x-scale \cdot \left(\left(a \cdot a\right) \cdot \left(x-scale \cdot 16\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\\ \end{array} \]

Alternative 17
Error	53.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -9.5 \cdot 10^{-142} \lor \neg \left(x-scale \leq 5.2 \cdot 10^{-32}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \end{array} \]

Alternative 18
Error	53.6
Cost	448

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

Reproduce

herbie shell --seed 2023073 
(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))))