Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.7 → 25.8

Time: 57.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2.9281638874020762 \cdot 10^{+153}:\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;a \leq -7.717914449197134 \cdot 10^{-106}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;a \leq -1.7355260372219387 \cdot 10^{-266}:\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;a \leq -1.3431085281301488 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{elif}\;a \leq 1.3145188603480247 \cdot 10^{-198} \lor \neg \left(a \leq 7.014308032463412 \cdot 10^{+135}\right):\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array}\]

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
 (if (<= a -2.9281638874020762e+153)
   (* -4.0 (/ (pow (* a b) 2.0) (* x-scale (* x-scale (* y-scale y-scale)))))
   (if (<= a -7.717914449197134e-106)
     (*
      -4.0
      (/ (* (* b b) (* a a)) (* (* x-scale y-scale) (* x-scale y-scale))))
     (if (<= a -1.7355260372219387e-266)
       (*
        -4.0
        (/ (pow (* a b) 2.0) (* x-scale (* x-scale (* y-scale y-scale)))))
       (if (<= a -1.3431085281301488e-294)
         (-
          (*
           (/
            (/
             (*
              (*
               (* 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)))
         (if (or (<= a 1.3145188603480247e-198)
                 (not (<= a 7.014308032463412e+135)))
           (*
            -4.0
            (/ (pow (* a b) 2.0) (* x-scale (* x-scale (* y-scale y-scale)))))
           (*
            -4.0
            (/
             (* (* b b) (* a a))
             (* (* x-scale y-scale) (* x-scale y-scale))))))))))

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 (a <= -2.9281638874020762e+153) {
		tmp = -4.0 * (pow((a * b), 2.0) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale))));
	} else if (a <= -7.717914449197134e-106) {
		tmp = -4.0 * (((b * b) * (a * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else if (a <= -1.7355260372219387e-266) {
		tmp = -4.0 * (pow((a * b), 2.0) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale))));
	} else if (a <= -1.3431085281301488e-294) {
		tmp = ((((((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));
	} else if ((a <= 1.3145188603480247e-198) || !(a <= 7.014308032463412e+135)) {
		tmp = -4.0 * (pow((a * b), 2.0) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale))));
	} else {
		tmp = -4.0 * (((b * b) * (a * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	}
	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 < -2.9281638874020762e153 or -7.7179144491971342e-106 < a < -1.73552603722193871e-266 or -1.3431085281301488e-294 < a < 1.3145188603480247e-198 or 7.0143080324634123e135 < a

   1. Initial program 43.2

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

   2. Taylor expanded around 0 44.8

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

   3. Simplified 42.4

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

   5. Applied pow2_binary64_159 42.4

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

   6. Applied pow2_binary64_159 42.4

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

   7. Applied pow-prod-down_binary64_149 27.1

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

   if -2.9281638874020762e153 < a < -7.7179144491971342e-106 or 1.3145188603480247e-198 < a < 7.0143080324634123e135

   1. Initial program 39.0

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

   2. Taylor expanded around 0 33.5

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

   3. Simplified 30.3

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

   5. Applied add-sqr-sqrt_binary64_100 30.3

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

   6. Simplified 30.3

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

   7. Simplified 24.1

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

   if -1.73552603722193871e-266 < a < -1.3431085281301488e-294

   1. Initial program 30.9

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\]
3. Recombined 3 regimes into one program.

4. Final simplification 25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9281638874020762 \cdot 10^{+153}:\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;a \leq -7.717914449197134 \cdot 10^{-106}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;a \leq -1.7355260372219387 \cdot 10^{-266}:\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;a \leq -1.3431085281301488 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{elif}\;a \leq 1.3145188603480247 \cdot 10^{-198} \lor \neg \left(a \leq 7.014308032463412 \cdot 10^{+135}\right):\\ \;\;\;\;-4 \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(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))))