Average Error: 41.2 → 37.9
Time: 1.0min
Precision: binary64
\[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\]
↓
\[\begin{array}{l} \mathbf{if}\;x-scale \leq -6.821398003204381 \cdot 10^{-145}:\\ \;\;\;\;-\frac{8 \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}^{2}} + \left(4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}} + 4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}}\right)}{{x-scale}^{2}}\\ \mathbf{elif}\;x-scale \leq 2.3598969122018188 \cdot 10^{-119}:\\ \;\;\;\;\frac{x-scale \cdot \frac{\frac{\frac{4 \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{x-scale}}{x-scale}}{y-scale} + y-scale \cdot \left(-4 \cdot \left(\frac{{\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{x-scale} \cdot \frac{{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}\right)\right)}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-{a}^{2} \cdot \left(4 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}} + \left(8 \cdot \frac{{\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)}{{y-scale}^{2} \cdot {x-scale}^{2}} + 4 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}}\right)\right)\\ \end{array}\]
\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}
\mathbf{if}\;x-scale \leq -6.821398003204381 \cdot 10^{-145}:\\
\;\;\;\;-\frac{8 \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}^{2}} + \left(4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}} + 4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}}\right)}{{x-scale}^{2}}\\

\mathbf{elif}\;x-scale \leq 2.3598969122018188 \cdot 10^{-119}:\\
\;\;\;\;\frac{x-scale \cdot \frac{\frac{\frac{4 \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{x-scale}}{x-scale}}{y-scale} + y-scale \cdot \left(-4 \cdot \left(\frac{{\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{x-scale} \cdot \frac{{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}\right)\right)}{x-scale \cdot y-scale}\\

\mathbf{else}:\\
\;\;\;\;-{a}^{2} \cdot \left(4 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}} + \left(8 \cdot \frac{{\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)}{{y-scale}^{2} \cdot {x-scale}^{2}} + 4 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}}\right)\right)\\

\end{array}
(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
 (if (<= x-scale -6.821398003204381e-145)
   (-
    (/
     (+
      (*
       8.0
       (/
        (*
         (pow a 2.0)
         (*
          (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0)
          (*
           (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)
           (pow b 2.0))))
        (pow y-scale 2.0)))
      (+
       (*
        4.0
        (/
         (*
          (pow a 2.0)
          (*
           (pow b 2.0)
           (pow (sin (* 0.005555555555555556 (* angle PI))) 4.0)))
         (pow y-scale 2.0)))
       (*
        4.0
        (/
         (*
          (pow a 2.0)
          (*
           (pow b 2.0)
           (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0)))
         (pow y-scale 2.0)))))
     (pow x-scale 2.0)))
   (if (<= x-scale 2.3598969122018188e-119)
     (/
      (+
       (*
        x-scale
        (/
         (/
          (/
           (*
            4.0
            (*
             (*
              (- (* b b) (* a a))
              (* (sin (* PI (/ angle 180.0))) (cos (* PI (/ angle 180.0)))))
             (*
              (- (* b b) (* a a))
              (* (sin (* PI (/ angle 180.0))) (cos (* PI (/ angle 180.0)))))))
           x-scale)
          x-scale)
         y-scale))
       (*
        y-scale
        (*
         -4.0
         (*
          (/
           (+
            (pow (* a (sin (* PI (/ angle 180.0)))) 2.0)
            (pow (* b (cos (* PI (/ angle 180.0)))) 2.0))
           x-scale)
          (/
           (+
            (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
            (pow (* b (sin (* PI (/ angle 180.0)))) 2.0))
           (* y-scale y-scale))))))
      (* x-scale y-scale))
     (-
      (*
       (pow a 2.0)
       (+
        (*
         4.0
         (/
          (* (pow b 2.0) (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0))
          (* (pow y-scale 2.0) (pow x-scale 2.0))))
        (+
         (*
          8.0
          (/
           (*
            (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0)
            (*
             (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)
             (pow b 2.0)))
           (* (pow y-scale 2.0) (pow x-scale 2.0))))
         (*
          4.0
          (/
           (*
            (pow b 2.0)
            (pow (sin (* 0.005555555555555556 (* angle PI))) 4.0))
           (* (pow y-scale 2.0) (pow x-scale 2.0)))))))))))
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 tmp;
	if (x_45_scale <= -6.821398003204381e-145) {
		tmp = -(((8.0 * ((pow(a, 2.0) * (pow(sin(0.005555555555555556 * (angle * ((double) M_PI))), 2.0) * (pow(cos(0.005555555555555556 * (angle * ((double) M_PI))), 2.0) * pow(b, 2.0)))) / pow(y_45_scale, 2.0))) + ((4.0 * ((pow(a, 2.0) * (pow(b, 2.0) * pow(sin(0.005555555555555556 * (angle * ((double) M_PI))), 4.0))) / pow(y_45_scale, 2.0))) + (4.0 * ((pow(a, 2.0) * (pow(b, 2.0) * pow(cos(0.005555555555555556 * (angle * ((double) M_PI))), 4.0))) / pow(y_45_scale, 2.0))))) / pow(x_45_scale, 2.0));
	} else if (x_45_scale <= 2.3598969122018188e-119) {
		tmp = ((x_45_scale * ((((4.0 * ((((b * b) - (a * a)) * (sin(((double) M_PI) * (angle / 180.0)) * cos(((double) M_PI) * (angle / 180.0)))) * (((b * b) - (a * a)) * (sin(((double) M_PI) * (angle / 180.0)) * cos(((double) M_PI) * (angle / 180.0)))))) / x_45_scale) / x_45_scale) / y_45_scale)) + (y_45_scale * (-4.0 * (((pow((a * sin(((double) M_PI) * (angle / 180.0))), 2.0) + pow((b * cos(((double) M_PI) * (angle / 180.0))), 2.0)) / x_45_scale) * ((pow((a * cos(((double) M_PI) * (angle / 180.0))), 2.0) + pow((b * sin(((double) M_PI) * (angle / 180.0))), 2.0)) / (y_45_scale * y_45_scale)))))) / (x_45_scale * y_45_scale);
	} else {
		tmp = -(pow(a, 2.0) * ((4.0 * ((pow(b, 2.0) * pow(cos(0.005555555555555556 * (angle * ((double) M_PI))), 4.0)) / (pow(y_45_scale, 2.0) * pow(x_45_scale, 2.0)))) + ((8.0 * ((pow(sin(0.005555555555555556 * (angle * ((double) M_PI))), 2.0) * (pow(cos(0.005555555555555556 * (angle * ((double) M_PI))), 2.0) * pow(b, 2.0))) / (pow(y_45_scale, 2.0) * pow(x_45_scale, 2.0)))) + (4.0 * ((pow(b, 2.0) * pow(sin(0.005555555555555556 * (angle * ((double) M_PI))), 4.0)) / (pow(y_45_scale, 2.0) * pow(x_45_scale, 2.0)))))));
	}
	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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x-scale < -6.8213980032043814e-145

    1. Initial program 39.3

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

    2. Taylor expanded around 0 35.5

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

    if -6.8213980032043814e-145 < x-scale < 2.35989691220181881e-119

    1. Initial program 50.9

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

    3. Applied associate-*r/_binary64_2050.9

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

    4. Applied associate-*l/_binary64_2148.9

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

    5. Applied associate-*l/_binary64_2149.0

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

    6. Applied frac-sub_binary64_8749.0

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

    7. Simplified49.2

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

    8. Simplified49.2

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

    if 2.35989691220181881e-119 < x-scale

    1. Initial program 38.8

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

    2. Taylor expanded around 0 35.2

      \[\leadsto \color{blue}{-1 \cdot \left(\left(4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \left(8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \cdot {a}^{2}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification37.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -6.821398003204381 \cdot 10^{-145}:\\ \;\;\;\;-\frac{8 \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}^{2}} + \left(4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}} + 4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{y-scale}^{2}}\right)}{{x-scale}^{2}}\\ \mathbf{elif}\;x-scale \leq 2.3598969122018188 \cdot 10^{-119}:\\ \;\;\;\;\frac{x-scale \cdot \frac{\frac{\frac{4 \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{x-scale}}{x-scale}}{y-scale} + y-scale \cdot \left(-4 \cdot \left(\frac{{\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{x-scale} \cdot \frac{{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}\right)\right)}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-{a}^{2} \cdot \left(4 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}} + \left(8 \cdot \frac{{\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)}{{y-scale}^{2} \cdot {x-scale}^{2}} + 4 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2} \cdot {x-scale}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021013 
(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))))