Average Error: 48.0 → 3.8
Time: 48.5s
Precision: binary64
\[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
↓
\[\begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;\frac{angle}{180} \leq -8.107673630762209 \cdot 10^{+244}:\\ \;\;\;\;\frac{-4 \cdot \left(\left|\frac{a}{y-scale} \cdot b\right| \cdot \left|t_0\right|\right)}{\left|x-scale\right|}\\ \mathbf{elif}\;\frac{angle}{180} \leq -0.00033619766571942186:\\ \;\;\;\;-4 \cdot {\left(\left|\frac{a \cdot b}{y-scale \cdot x-scale}\right|\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]
\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}
t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\
\mathbf{if}\;\frac{angle}{180} \leq -8.107673630762209 \cdot 10^{+244}:\\
\;\;\;\;\frac{-4 \cdot \left(\left|\frac{a}{y-scale} \cdot b\right| \cdot \left|t_0\right|\right)}{\left|x-scale\right|}\\

\mathbf{elif}\;\frac{angle}{180} \leq -0.00033619766571942186:\\
\;\;\;\;-4 \cdot {\left(\left|\frac{a \cdot b}{y-scale \cdot x-scale}\right|\right)}^{2}\\

\mathbf{else}:\\
\;\;\;\;-4 \cdot \left(t_0 \cdot t_0\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
 (let* ((t_0 (* (/ a y-scale) (/ b x-scale))))
   (if (<= (/ angle 180.0) -8.107673630762209e+244)
     (/ (* -4.0 (* (fabs (* (/ a y-scale) b)) (fabs t_0))) (fabs x-scale))
     (if (<= (/ angle 180.0) -0.00033619766571942186)
       (* -4.0 (pow (fabs (/ (* a b) (* y-scale x-scale))) 2.0))
       (* -4.0 (* t_0 t_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 t_0 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if ((angle / 180.0) <= -8.107673630762209e+244) {
		tmp = (-4.0 * (fabs((a / y_45_scale) * b) * fabs(t_0))) / fabs(x_45_scale);
	} else if ((angle / 180.0) <= -0.00033619766571942186) {
		tmp = -4.0 * pow(fabs((a * b) / (y_45_scale * x_45_scale)), 2.0);
	} else {
		tmp = -4.0 * (t_0 * t_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 (/.f64 angle 180) < -8.107673630762209e244

    1. Initial program 49.7

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

    2. Taylor expanded in angle around 0 30.5

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

    3. Applied pow-prod-down_binary6424.1

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

    4. Applied add-sqr-sqrt_binary6424.1

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

    5. Simplified24.2

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

    6. Simplified3.4

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

    7. Applied associate-*r/_binary644.4

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

    8. Applied fabs-div_binary644.4

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

    9. Applied associate-*l/_binary645.2

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

    10. Applied associate-*r/_binary645.2

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

    if -8.107673630762209e244 < (/.f64 angle 180) < -3.3619766571942186e-4

    1. Initial program 50.5

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

    2. Taylor expanded in angle around 0 33.7

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

    3. Applied pow-prod-down_binary6424.7

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

    4. Applied add-sqr-sqrt_binary6424.7

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

    5. Simplified25.5

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

    6. Simplified4.3

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

    7. Taylor expanded in a around 0 3.8

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

    if -3.3619766571942186e-4 < (/.f64 angle 180)

    1. Initial program 47.2

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

    2. Taylor expanded in angle around 0 34.0

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

    3. Applied pow-prod-down_binary6426.1

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

    4. Applied add-sqr-sqrt_binary6426.1

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

    5. Simplified26.6

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

    6. Simplified3.7

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

    7. Applied sqr-abs_binary643.7

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -8.107673630762209 \cdot 10^{+244}:\\ \;\;\;\;\frac{-4 \cdot \left(\left|\frac{a}{y-scale} \cdot b\right| \cdot \left|\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right|\right)}{\left|x-scale\right|}\\ \mathbf{elif}\;\frac{angle}{180} \leq -0.00033619766571942186:\\ \;\;\;\;-4 \cdot {\left(\left|\frac{a \cdot b}{y-scale \cdot x-scale}\right|\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))