Average Error: 40.8 → 5.8
Time: 56.8s
Precision: binary64
\[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\]
↓
\[\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)\]
\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}
↓
\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)
(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
 (*
  (fabs (* (/ a x-scale) (/ b y-scale)))
  (* (fabs (* (/ a x-scale) (/ b y-scale))) -4.0)))
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) {
	return fabs((a / x_45_scale) * (b / y_45_scale)) * (fabs((a / x_45_scale) * (b / y_45_scale)) * -4.0);
}

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. Initial program 40.8

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

  2. Taylor expanded around 0 39.1

    \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\]

  3. Simplified36.3

    \[\leadsto \color{blue}{-4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}\]
  4. Using strategy rm

  5. Applied pow2_binary64_15936.3

    \[\leadsto -4 \cdot \frac{\left(b \cdot b\right) \cdot \color{blue}{{a}^{2}}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\]

  6. Applied pow2_binary64_15936.3

    \[\leadsto -4 \cdot \frac{\color{blue}{{b}^{2}} \cdot {a}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\]

  7. Applied pow-prod-down_binary64_14927.3

    \[\leadsto -4 \cdot \frac{\color{blue}{{\left(b \cdot a\right)}^{2}}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt_binary64_10027.3

    \[\leadsto -4 \cdot \color{blue}{\left(\sqrt{\frac{{\left(b \cdot a\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \cdot \sqrt{\frac{{\left(b \cdot a\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}\right)}\]

  10. Simplified28.8

    \[\leadsto -4 \cdot \left(\color{blue}{\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|} \cdot \sqrt{\frac{{\left(b \cdot a\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}\right)\]

  11. Simplified5.8

    \[\leadsto -4 \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \color{blue}{\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|}\right)\]
  12. Using strategy rm

  13. Applied associate-*r*_binary64_185.8

    \[\leadsto \color{blue}{\left(-4 \cdot \left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|\right) \cdot \left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|}\]

  14. Simplified5.8

    \[\leadsto \color{blue}{\left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)} \cdot \left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|\]

  15. Final simplification5.8

    \[\leadsto \left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)\]

Reproduce

herbie shell --seed 2020346 
(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))))