Average Error: 0.5 → 0.4
Time: 14.5s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\cos x \cdot \left(1.5 + 1.5 \cdot \sqrt{5}\right)}\right)}\]
\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 + \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 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\cos x \cdot \left(1.5 + 1.5 \cdot \sqrt{5}\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 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (+
   3.0
   (+
    (* (cos y) (* 1.5 (/ 4.0 (+ 3.0 (sqrt 5.0)))))
    (/
     (- (* (* (cos x) (cos x)) 11.25) (* (cos x) (* (cos x) 2.25)))
     (* (cos x) (+ 1.5 (* 1.5 (sqrt 5.0)))))))))
double code(double x, double y) {
	return (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))));
}
↓
double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((cos(y) * (1.5 * (4.0 / (3.0 + sqrt(5.0))))) + ((((cos(x) * cos(x)) * 11.25) - (cos(x) * (cos(x) * 2.25))) / (cos(x) * (1.5 + (1.5 * sqrt(5.0)))))));
}

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 0.5

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}}\]
  3. Using strategy rm

  4. Applied sub-neg_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)\right)}\]

  5. Applied distribute-rgt-in_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \color{blue}{\left(\sqrt{5} \cdot \cos x + \left(-1\right) \cdot \cos x\right)}\right)}\]

  6. Applied distribute-rgt-in_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\left(\left(\sqrt{5} \cdot \cos x\right) \cdot 1.5 + \left(\left(-1\right) \cdot \cos x\right) \cdot 1.5\right)}\right)}\]

  7. Simplified0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \left(\color{blue}{1.5 \cdot \left(\cos x \cdot \sqrt{5}\right)} + \left(\left(-1\right) \cdot \cos x\right) \cdot 1.5\right)\right)}\]

  8. Simplified0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \left(1.5 \cdot \left(\cos x \cdot \sqrt{5}\right) + \color{blue}{\cos x \cdot -1.5}\right)\right)}\]
  9. Using strategy rm

  10. Applied flip-+_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{\left(1.5 \cdot \left(\cos x \cdot \sqrt{5}\right)\right) \cdot \left(1.5 \cdot \left(\cos x \cdot \sqrt{5}\right)\right) - \left(\cos x \cdot -1.5\right) \cdot \left(\cos x \cdot -1.5\right)}{1.5 \cdot \left(\cos x \cdot \sqrt{5}\right) - \cos x \cdot -1.5}}\right)}\]

  11. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \frac{\color{blue}{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}}{1.5 \cdot \left(\cos x \cdot \sqrt{5}\right) - \cos x \cdot -1.5}\right)}\]

  12. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\color{blue}{\cos x \cdot \left(\sqrt{5} \cdot 1.5 + 1.5\right)}}\right)}\]
  13. Using strategy rm

  14. Applied flip--_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\cos x \cdot \left(\sqrt{5} \cdot 1.5 + 1.5\right)}\right)}\]

  15. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\cos x \cdot \left(\sqrt{5} \cdot 1.5 + 1.5\right)}\right)}\]

  16. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{\left(\cos x \cdot \cos x\right) \cdot 11.25 - \cos x \cdot \left(\cos x \cdot 2.25\right)}{\cos x \cdot \left(1.5 + 1.5 \cdot \sqrt{5}\right)}\right)}\]

Reproduce

herbie shell --seed 2020268 
(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))))))