Average Error: 0.5 → 0.4
Time: 14.4s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \sqrt{\sqrt{2}} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \sqrt{\sqrt{2}} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}
(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))
↓
(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt (sqrt 2.0))
    (*
     (- (sin x) (/ (sin y) 16.0))
     (*
      (sqrt (sqrt 2.0))
      (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))))))
  (*
   3.0
   (+
    1.0
    (+
     (* (cos x) (/ (- (sqrt 5.0) 1.0) 2.0))
     (* (cos y) (/ (/ (- (* 3.0 3.0) 5.0) (+ 3.0 (sqrt 5.0))) 2.0)))))))
double code(double x, double y) {
	return (((double) (2.0 + ((double) (((double) (((double) (((double) sqrt(2.0)) * ((double) (((double) sin(x)) - (((double) sin(y)) / 16.0))))) * ((double) (((double) sin(y)) - (((double) sin(x)) / 16.0))))) * ((double) (((double) cos(x)) - ((double) cos(y)))))))) / ((double) (3.0 * ((double) (((double) (1.0 + ((double) ((((double) (((double) sqrt(5.0)) - 1.0)) / 2.0) * ((double) cos(x)))))) + ((double) ((((double) (3.0 - ((double) sqrt(5.0)))) / 2.0) * ((double) cos(y)))))))));
}
↓
double code(double x, double y) {
	return (((double) (2.0 + ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) (((double) sin(x)) - (((double) sin(y)) / 16.0))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) (((double) cos(x)) - ((double) cos(y)))) * ((double) (((double) sin(y)) - (((double) sin(x)) / 16.0))))))))))))) / ((double) (3.0 * ((double) (1.0 + ((double) (((double) (((double) cos(x)) * (((double) (((double) sqrt(5.0)) - 1.0)) / 2.0))) + ((double) (((double) cos(y)) * ((((double) (((double) (3.0 * 3.0)) - 5.0)) / ((double) (3.0 + ((double) sqrt(5.0))))) / 2.0))))))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 0.5 bits

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  2. SimplifiedError: 0.5 bits

    \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)}}\]
  3. Using strategy rm

  4. Applied flip--Error: 0.5 bits

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2}\right)\right)}\]

  5. SimplifiedError: 0.4 bits

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{\color{blue}{3 \cdot 3 - 5}}{3 + \sqrt{5}}}{2}\right)\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrtError: 0.4 bits

    \[\leadsto \frac{2 + \sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  8. Applied sqrt-prodError: 0.4 bits

    \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  9. Applied associate-*l*Error: 0.4 bits

    \[\leadsto \frac{2 + \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)\right)}}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  10. SimplifiedError: 0.4 bits

    \[\leadsto \frac{2 + \sqrt{\sqrt{2}} \cdot \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \sqrt{\sqrt{2}}\right)\right)}}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  11. Final simplificationError: 0.4 bits

    \[\leadsto \frac{2 + \sqrt{\sqrt{2}} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))