b from scale-rotated-ellipse

Average Error: 64.0 → 43.3

Time: 2.3min

Precision: binary64

Math

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

↓

\[-0.25 \cdot \log \left({\left(e^{\sqrt[3]{0}}\right)}^{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (-
      (+
       (/
        (/
         (+
          (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))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (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))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 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)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -0.25 (log (pow (exp (cbrt 0.0)) (pow (* x-scale y-scale) 2.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt(((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) - sqrt(pow(((((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)), 2.0) + pow((((((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)))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -0.25 * log(pow(exp(cbrt(0.0)), pow((x_45_scale * y_45_scale), 2.0)));
}

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. Initial program 64.0

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

2. Simplified 63.9

   \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right)} \]

3. Taylor expanded around 0 63.6

   \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \color{blue}{\frac{\left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right) - \sqrt{{a}^{4} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + \left(2 \cdot \left({a}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {b}^{4}\right)}}{{x-scale}^{2}}}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]

4. Simplified 63.6

   \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \color{blue}{\frac{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right) - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {a}^{4} + \left({b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)\right)}}{x-scale \cdot x-scale}}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]

5. Taylor expanded around 0 61.6

   \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{\color{blue}{0}}{x-scale \cdot x-scale}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]

6. Simplified 62.7

   \[\leadsto -0.25 \cdot \left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{\color{blue}{{angle}^{4} \cdot 0}}{x-scale \cdot x-scale}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]
7. Using strategy rm

8. Applied add-cbrt-cube_binary64 62.7

   \[\leadsto -0.25 \cdot \left(\color{blue}{\sqrt[3]{\left(\frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{{angle}^{4} \cdot 0}{x-scale \cdot x-scale}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{{angle}^{4} \cdot 0}{x-scale \cdot x-scale}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)}\right) \cdot \frac{\sqrt{\left(\frac{8}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{{angle}^{4} \cdot 0}{x-scale \cdot x-scale}\right)}}{\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)}}} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]

9. Simplified 44.4

   \[\leadsto -0.25 \cdot \left(\sqrt[3]{\color{blue}{0}} \cdot {\left(x-scale \cdot y-scale\right)}^{2}\right) \]
10. Using strategy rm

11. Applied add-log-exp_binary64 44.4

    \[\leadsto -0.25 \cdot \color{blue}{\log \left(e^{\sqrt[3]{0} \cdot {\left(x-scale \cdot y-scale\right)}^{2}}\right)} \]

12. Simplified 43.3

    \[\leadsto -0.25 \cdot \log \color{blue}{\left({\left(e^{\sqrt[3]{0}}\right)}^{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}\right)} \]

13. Final simplification 43.3

    \[\leadsto -0.25 \cdot \log \left({\left(e^{\sqrt[3]{0}}\right)}^{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}\right) \]

Reproduce

herbie shell --seed 2021190 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))