Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.1 → 6.0

Time: 56.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 3.833516390112431 \cdot 10^{-117}:\\ \;\;\;\;\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 2.598555315437982 \cdot 10^{+79}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale} \cdot \frac{a \cdot b}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|\right)}^{2}\\ \end{array}\]

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
 (if (<= y-scale 3.833516390112431e-117)
   (*
    (fabs (* (/ a x-scale) (/ b y-scale)))
    (* (fabs (* (/ a x-scale) (/ b y-scale))) -4.0))
   (if (<= y-scale 2.598555315437982e+79)
     (*
      -4.0
      (* (/ (* a b) x-scale) (/ (* a b) (* x-scale (* y-scale y-scale)))))
     (* -4.0 (pow (fabs (* (/ a x-scale) (/ b y-scale))) 2.0)))))

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 tmp;
	if (y_45_scale <= 3.833516390112431e-117) {
		tmp = fabs((a / x_45_scale) * (b / y_45_scale)) * (fabs((a / x_45_scale) * (b / y_45_scale)) * -4.0);
	} else if (y_45_scale <= 2.598555315437982e+79) {
		tmp = -4.0 * (((a * b) / x_45_scale) * ((a * b) / (x_45_scale * (y_45_scale * y_45_scale))));
	} else {
		tmp = -4.0 * pow(fabs((a / x_45_scale) * (b / y_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

Derivation

1. Split input into 3 regimes

2. if y-scale < 3.8335163901124309e-117

   1. Initial program 42.9

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

   2. Taylor expanded around 0 42.5

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

   3. Simplified 39.9

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

   5. Applied pow2_binary64_159 39.9

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

   6. Applied pow2_binary64_159 39.9

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

   7. Applied pow-prod-down_binary64_149 31.6

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

   9. Applied add-sqr-sqrt_binary64_100 31.6

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

   10. Simplified 33.0

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

   11. Simplified 6.4

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

   13. Applied associate-*r*_binary64_18 6.4

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

   14. Simplified 6.4

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

   if 3.8335163901124309e-117 < y-scale < 2.598555315437982e79

   1. Initial program 40.6

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

   2. Taylor expanded around 0 33.9

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

   3. Simplified 32.8

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

   5. Applied pow2_binary64_159 32.8

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

   6. Applied pow2_binary64_159 32.8

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

   7. Applied pow-prod-down_binary64_149 22.1

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

   9. Applied unpow2_binary64_143 22.1

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

   10. Applied times-frac_binary64_84 5.1

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

   if 2.598555315437982e79 < y-scale

   1. Initial program 37.0

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

   2. Taylor expanded around 0 35.5

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

   3. Simplified 31.2

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

   5. Applied pow2_binary64_159 31.2

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

   6. Applied pow2_binary64_159 31.2

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

   7. Applied pow-prod-down_binary64_149 21.6

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

   9. Applied add-sqr-sqrt_binary64_100 21.6

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

   10. Simplified 22.8

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

   11. Simplified 5.8

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

   13. Applied pow2_binary64_159 5.8

       \[\leadsto -4 \cdot \color{blue}{{\left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|\right)}^{2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.833516390112431 \cdot 10^{-117}:\\ \;\;\;\;\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot \left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right| \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 2.598555315437982 \cdot 10^{+79}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale} \cdot \frac{a \cdot b}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\left|\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right|\right)}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))))