Average Error: 41.1 → 5.7
Time: 1.9min
Precision: binary64
\[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\]
↓
\[-4 \cdot \frac{\frac{b \cdot a}{\left|x-scale \cdot y-scale\right|}}{\frac{\left|x-scale \cdot y-scale\right|}{b \cdot a}}\]
\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}
↓
-4 \cdot \frac{\frac{b \cdot a}{\left|x-scale \cdot y-scale\right|}}{\frac{\left|x-scale \cdot y-scale\right|}{b \cdot a}}
(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
 (*
  -4.0
  (/
   (/ (* b a) (fabs (* x-scale y-scale)))
   (/ (fabs (* x-scale y-scale)) (* b a)))))
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 -4.0 * (((b * a) / fabs(x_45_scale * y_45_scale)) / (fabs(x_45_scale * y_45_scale) / (b * a)));
}

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 41.1

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

  2. Taylor expanded around 0 39.8

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

  3. Simplified36.9

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

  5. Applied clear-num_binary64_7736.9

    \[\leadsto -4 \cdot \color{blue}{\frac{1}{\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}}}\]

  6. Simplified27.5

    \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt_binary64_10027.5

    \[\leadsto -4 \cdot \frac{1}{\frac{\color{blue}{\sqrt{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \sqrt{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}}\]

  9. Simplified27.5

    \[\leadsto -4 \cdot \frac{1}{\frac{\color{blue}{\left|x-scale \cdot y-scale\right|} \cdot \sqrt{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}}\]

  10. Simplified19.5

    \[\leadsto -4 \cdot \frac{1}{\frac{\left|x-scale \cdot y-scale\right| \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}}\]
  11. Using strategy rm

  12. Applied times-frac_binary64_845.9

    \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{\left|x-scale \cdot y-scale\right|}{b \cdot a} \cdot \frac{\left|x-scale \cdot y-scale\right|}{b \cdot a}}}\]

  13. Applied associate-/r*_binary64_225.7

    \[\leadsto -4 \cdot \color{blue}{\frac{\frac{1}{\frac{\left|x-scale \cdot y-scale\right|}{b \cdot a}}}{\frac{\left|x-scale \cdot y-scale\right|}{b \cdot a}}}\]

  14. Simplified5.7

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

  15. Final simplification5.7

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

Reproduce

herbie shell --seed 2021007 
(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))))