Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.4 → 29.6

Time: 1.6min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t_0\\ t_2 := \left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\\ t_3 := \frac{\left(b \cdot b\right) \cdot {t_1}^{4}}{t_2}\\ t_4 := \cos t_0\\ t_5 := \left(b \cdot b\right) \cdot {t_4}^{4}\\ t_6 := 8 \cdot \frac{{t_4}^{2} \cdot \left(\left(b \cdot b\right) \cdot {t_1}^{2}\right)}{t_2}\\ \mathbf{if}\;a \leq -5.291857281523912 \cdot 10^{-67}:\\ \;\;\;\;-a \cdot \left(a \cdot \mathsf{fma}\left(4, t_3, \mathsf{fma}\left(4, \frac{\frac{t_5}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}, t_6\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.7136344850281534 \cdot 10^{+56}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(a \cdot \mathsf{fma}\left(4, t_3, \mathsf{fma}\left(4, \frac{t_5}{{\left(y-scale \cdot x-scale\right)}^{2}}, t_6\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* (* y-scale y-scale) (* x-scale x-scale)))
        (t_3 (/ (* (* b b) (pow t_1 4.0)) t_2))
        (t_4 (cos t_0))
        (t_5 (* (* b b) (pow t_4 4.0)))
        (t_6 (* 8.0 (/ (* (pow t_4 2.0) (* (* b b) (pow t_1 2.0))) t_2))))
   (if (<= a -5.291857281523912e-67)
     (-
      (*
       a
       (*
        a
        (fma
         4.0
         t_3
         (fma 4.0 (/ (/ t_5 (* y-scale y-scale)) (* x-scale x-scale)) t_6)))))
     (if (<= a 2.7136344850281534e+56)
       0.0
       (-
        (*
         a
         (*
          a
          (fma
           4.0
           t_3
           (fma 4.0 (/ t_5 (pow (* y-scale x-scale) 2.0)) t_6)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((((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 * (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)) - ((4.0 * (((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));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = (y_45_scale * y_45_scale) * (x_45_scale * x_45_scale);
	double t_3 = ((b * b) * pow(t_1, 4.0)) / t_2;
	double t_4 = cos(t_0);
	double t_5 = (b * b) * pow(t_4, 4.0);
	double t_6 = 8.0 * ((pow(t_4, 2.0) * ((b * b) * pow(t_1, 2.0))) / t_2);
	double tmp;
	if (a <= -5.291857281523912e-67) {
		tmp = -(a * (a * fma(4.0, t_3, fma(4.0, ((t_5 / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale)), t_6))));
	} else if (a <= 2.7136344850281534e+56) {
		tmp = 0.0;
	} else {
		tmp = -(a * (a * fma(4.0, t_3, fma(4.0, (t_5 / pow((y_45_scale * x_45_scale), 2.0)), t_6))));
	}
	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 a < -5.291857281523912e-67

   1. Initial program 49.3

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

   2. Taylor expanded in a around 0 43.9

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

   3. Simplified 43.9

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

   4. Applied associate-*l*_binary64 36.8

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

   5. Applied associate-/r*_binary64 37.0

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

   if -5.291857281523912e-67 < a < 2.7136344850281534e56

   1. Initial program 34.7

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

   2. Taylor expanded in b around 0 32.7

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

   3. Simplified 24.5

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

   if 2.7136344850281534e56 < a

   1. Initial program 57.1

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

   2. Taylor expanded in a around 0 51.0

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

   3. Simplified 51.0

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

   4. Applied associate-*l*_binary64 40.0

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

   5. Applied pow2_binary64 40.0

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

   6. Applied pow2_binary64 40.0

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

   7. Applied pow-prod-down_binary64 39.0

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

   8. Simplified 39.0

      \[\leadsto -a \cdot \left(a \cdot \mathsf{fma}\left(4, \frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}, \mathsf{fma}\left(4, \frac{\left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}}, 8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.291857281523912 \cdot 10^{-67}:\\ \;\;\;\;-a \cdot \left(a \cdot \mathsf{fma}\left(4, \frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}, \mathsf{fma}\left(4, \frac{\frac{\left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}, 8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.7136344850281534 \cdot 10^{+56}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(a \cdot \mathsf{fma}\left(4, \frac{\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}, \mathsf{fma}\left(4, \frac{\left(b \cdot b\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{\left(y-scale \cdot x-scale\right)}^{2}}, 8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 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 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (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))))