b from scale-rotated-ellipse

Average Error: 63.9 → 49.7

Time: 2.1min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.755251793459713 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2} \cdot x-scale} \cdot {\left(y-scale \cdot x-scale\right)}^{2}\right) \cdot 0.25\\ \mathbf{elif}\;y-scale \leq 6.879951835079887 \cdot 10^{-110}:\\ \;\;\;\;-0.25 \cdot \log \left({\left(e^{\frac{\sqrt{\frac{\left(a \cdot a\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot \left(\frac{8}{{\left(y-scale \cdot x-scale\right)}^{2}} \cdot {\left(a \cdot b\right)}^{4}\right)}}{\left(a \cdot b\right) \cdot \left(b \cdot \left(-a\right)\right)}}\right)}^{\left({\left(y-scale \cdot x-scale\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{8} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot x-scale\right)\right)\right) \cdot 0.25\\ \end{array}\]

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
 (if (<= y-scale -1.755251793459713e-50)
   (*
    (*
     (/
      (* a (* (sqrt 8.0) (cos (* 0.005555555555555556 (* angle PI)))))
      (* (pow y-scale 2.0) x-scale))
     (pow (* y-scale x-scale) 2.0))
    0.25)
   (if (<= y-scale 6.879951835079887e-110)
     (*
      -0.25
      (log
       (pow
        (exp
         (/
          (sqrt
           (*
            (/
             (* (* a a) (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0))
             (* y-scale y-scale))
            (* (/ 8.0 (pow (* y-scale x-scale) 2.0)) (pow (* a b) 4.0))))
          (* (* a b) (* b (- a)))))
        (pow (* y-scale x-scale) 2.0))))
     (*
      (*
       a
       (* (sqrt 8.0) (* (cos (* 0.005555555555555556 (* angle PI))) x-scale)))
      0.25))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= -1.755251793459713e-50) {
		tmp = (((a * (sqrt(8.0) * cos(0.005555555555555556 * (angle * ((double) M_PI))))) / (pow(y_45_scale, 2.0) * x_45_scale)) * pow((y_45_scale * x_45_scale), 2.0)) * 0.25;
	} else if (y_45_scale <= 6.879951835079887e-110) {
		tmp = -0.25 * log(pow(exp(sqrt((((a * a) * pow(cos(0.005555555555555556 * (angle * ((double) M_PI))), 2.0)) / (y_45_scale * y_45_scale)) * ((8.0 / pow((y_45_scale * x_45_scale), 2.0)) * pow((a * b), 4.0))) / ((a * b) * (b * -a))), pow((y_45_scale * x_45_scale), 2.0)));
	} else {
		tmp = (a * (sqrt(8.0) * (cos(0.005555555555555556 * (angle * ((double) M_PI))) * x_45_scale))) * 0.25;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Bits error versus x-scale

Bits error versus y-scale

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.7552517934597131e-50

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around inf 63.1

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

   4. Simplified 63.1

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

   6. Applied add-cube-cbrt_binary64 63.2

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \sqrt{\color{blue}{\left(\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}\right) \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   7. Applied sqrt-prod_binary64 63.2

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \color{blue}{\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}} \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   8. Applied cancel-sign-sub-inv_binary64 63.2

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \left(-\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right)}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   9. Simplified 63.2

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \color{blue}{\left(-\left|\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}\right|\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   10. Taylor expanded around inf 61.2

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

   11. Simplified 61.2

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

   12. Taylor expanded around 0 53.4

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

   if -1.7552517934597131e-50 < y-scale < 6.87995183507988736e-110

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around inf 64.0

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

   4. Simplified 64.0

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

   6. Applied add-cube-cbrt_binary64 64.0

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \sqrt{\color{blue}{\left(\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}\right) \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   7. Applied sqrt-prod_binary64 64.0

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \color{blue}{\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}} \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   8. Applied cancel-sign-sub-inv_binary64 64.0

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \left(-\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right)}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   9. Simplified 64.0

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \color{blue}{\left(-\left|\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}\right|\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   10. Taylor expanded around inf 63.5

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

   11. Simplified 63.5

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

   13. Applied add-log-exp_binary64 63.5

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

   14. Simplified 43.6

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

   if 6.87995183507988736e-110 < y-scale

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around inf 63.4

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

   4. Simplified 63.4

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

   6. Applied add-cube-cbrt_binary64 63.4

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \sqrt{\color{blue}{\left(\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}\right) \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   7. Applied sqrt-prod_binary64 63.4

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) - \color{blue}{\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}} \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   8. Applied cancel-sign-sub-inv_binary64 63.4

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \left(-\sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)} \cdot \sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4}}{{y-scale}^{4}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}{{y-scale}^{4}}\right)}}\right)}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   9. Simplified 63.4

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale}\right) + \color{blue}{\left(-\left|\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}\right|\right) \cdot \sqrt{\sqrt[3]{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}}{{y-scale}^{4}} + \left(\frac{{a}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} + 2 \cdot \frac{\left(a \cdot a\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)\right)}{{y-scale}^{4}}\right)}}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)\]

   10. Taylor expanded around inf 61.1

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

   11. Simplified 61.1

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

   12. Taylor expanded around 0 53.0

       \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \left(\sqrt{8} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot x-scale\right)\right)\right)\right)}\]

   13. Simplified 53.0

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

4. Final simplification 49.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -1.755251793459713 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2} \cdot x-scale} \cdot {\left(y-scale \cdot x-scale\right)}^{2}\right) \cdot 0.25\\ \mathbf{elif}\;y-scale \leq 6.879951835079887 \cdot 10^{-110}:\\ \;\;\;\;-0.25 \cdot \log \left({\left(e^{\frac{\sqrt{\frac{\left(a \cdot a\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot \left(\frac{8}{{\left(y-scale \cdot x-scale\right)}^{2}} \cdot {\left(a \cdot b\right)}^{4}\right)}}{\left(a \cdot b\right) \cdot \left(b \cdot \left(-a\right)\right)}}\right)}^{\left({\left(y-scale \cdot x-scale\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{8} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot x-scale\right)\right)\right) \cdot 0.25\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))