raw-angle from scale-rotated-ellipse

Average Error: 55.4 → 47.3

Time: 5.5min

Precision: binary64

Cost: 188482

Math

\[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

↓

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1.3672882093309752 \cdot 10^{+99}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.492788770752604 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {b}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sqrt[3]{{\sin \left(\frac{angle}{180} \cdot \pi\right)}^{3}}\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale -1.3672882093309752e+99)
   0.0
   (if (<= x-scale 1.492788770752604e-59)
     (*
      180.0
      (/
       (atan
        (/
         (-
          (-
           (/
            (/
             (+
              (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
              (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
             y-scale)
            y-scale)
           (/
            (/
             (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow b 2.0))
             x-scale)
            x-scale))
          (sqrt
           (+
            (pow
             (-
              (/
               (/
                (+
                 (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
                 (pow
                  (*
                   b
                   (-
                    1.0
                    (* 1.54320987654321e-05 (* (* angle angle) (pow PI 2.0)))))
                  2.0))
                x-scale)
               x-scale)
              (/
               (/
                (+ (pow (* b (sin (* (/ angle 180.0) PI))) 2.0) (pow a 2.0))
                y-scale)
               y-scale))
             2.0)
            (pow
             (/
              (/
               (*
                (cos (* (/ angle 180.0) PI))
                (*
                 (sin (* (/ angle 180.0) PI))
                 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
               x-scale)
              y-scale)
             2.0))))
         (/
          (/
           (*
            (cos (* (/ angle 180.0) PI))
            (*
             (* 2.0 (- (pow b 2.0) (pow a 2.0)))
             (cbrt (pow (sin (* (/ angle 180.0) PI)) 3.0))))
           x-scale)
          y-scale)))
       PI))
     0.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((((((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) - (((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)) - 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))) / (((((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)) / ((double) M_PI));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= -1.3672882093309752e+99) {
		tmp = 0.0;
	} else if (x_45_scale <= 1.492788770752604e-59) {
		tmp = 180.0 * (atan((((((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) - (((pow((a * sin((angle / 180.0) * ((double) M_PI))), 2.0) + pow(b, 2.0)) / x_45_scale) / x_45_scale)) - sqrt(pow(((((pow((a * sin((angle / 180.0) * ((double) M_PI))), 2.0) + pow((b * (1.0 - (1.54320987654321e-05 * ((angle * angle) * pow(((double) M_PI), 2.0))))), 2.0)) / x_45_scale) / x_45_scale) - (((pow((b * sin((angle / 180.0) * ((double) M_PI))), 2.0) + pow(a, 2.0)) / y_45_scale) / y_45_scale)), 2.0) + pow((((cos((angle / 180.0) * ((double) M_PI)) * (sin((angle / 180.0) * ((double) M_PI)) * (2.0 * (pow(b, 2.0) - pow(a, 2.0))))) / x_45_scale) / y_45_scale), 2.0))) / (((cos((angle / 180.0) * ((double) M_PI)) * ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * cbrt(pow(sin((angle / 180.0) * ((double) M_PI)), 3.0)))) / x_45_scale) / y_45_scale)) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	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

Alternatives

Alternative 1
Error	47.3
Cost	175618

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -137950.9144803258:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.162332324042035 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {b}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 2
Error	47.3
Cost	175618

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -42846317444.72637:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.2284236133841489 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {b}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 3
Error	47.4
Cost	143362

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -578248351.1093605:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.492788770752604 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 4
Error	47.7
Cost	143170

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -6110227.905919393:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 8.476878715525107 \cdot 10^{-60}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {b}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 5
Error	47.8
Cost	136386

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1340035485.5921035:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.492788770752604 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 6
Error	51.8
Cost	64

\[0\]

Alternative 7
Error	60.3
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if x-scale < -1.3672882093309752e99 or 1.49278877075260393e-59 < x-scale

   1. Initial program 43.1

      \[0\]

   2. Simplified 43.1

      \[\leadsto \color{blue}{0}\]

   if -1.3672882093309752e99 < x-scale < 1.49278877075260393e-59

   1. Initial program 52.2

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   2. Taylor expanded around 0 51.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   3. Simplified 51.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)}\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   4. Taylor expanded around 0 51.5

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   5. Taylor expanded around 0 51.5

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube_binary64 51.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   8. Simplified 51.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sqrt[3]{\color{blue}{{\sin \left(\frac{angle}{180} \cdot \pi\right)}^{3}}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}\]

   9. Simplified 51.0

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sqrt[3]{{\sin \left(\frac{angle}{180} \cdot \pi\right)}^{3}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.3672882093309752 \cdot 10^{+99}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 1.492788770752604 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {b}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(1 - 1.54320987654321 \cdot 10^{-05} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {a}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sqrt[3]{{\sin \left(\frac{angle}{180} \cdot \pi\right)}^{3}}\right)}{x-scale}}{y-scale}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(FPCore (a b angle x-scale y-scale)
  :name "raw-angle from scale-rotated-ellipse"
  :precision binary64
  (* 180.0 (/ (atan (/ (- (- (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale) (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-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)))) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale))) PI)))