Average Error: 0.5 → 0.4
Time: 11.0s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{1.5 \cdot \left(\cos x \cdot \left(-1 + 5 \cdot \sqrt{5}\right)\right)}{\sqrt{5} \cdot \sqrt{5} + \left(\sqrt{5} + 1\right)}\right)}\]
\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 \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{1.5 \cdot \left(\cos x \cdot \left(-1 + 5 \cdot \sqrt{5}\right)\right)}{\sqrt{5} \cdot \sqrt{5} + \left(\sqrt{5} + 1\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)))
    (/
     (- (pow (cos x) 3.0) (pow (cos y) 3.0))
     (+ (* (cos x) (cos x)) (* (cos y) (+ (cos x) (cos y)))))))
  (+
   3.0
   (+
    (* (cos y) (* 1.5 (/ 4.0 (+ 3.0 (sqrt 5.0)))))
    (/
     (* 1.5 (* (cos x) (+ -1.0 (* 5.0 (sqrt 5.0)))))
     (+ (* (sqrt 5.0) (sqrt 5.0)) (+ (sqrt 5.0) 1.0)))))))
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))) * ((pow(cos(x), 3.0) - pow(cos(y), 3.0)) / ((cos(x) * cos(x)) + (cos(y) * (cos(x) + cos(y))))))) / (3.0 + ((cos(y) * (1.5 * (4.0 / (3.0 + sqrt(5.0))))) + ((1.5 * (cos(x) * (-1.0 + (5.0 * sqrt(5.0))))) / ((sqrt(5.0) * sqrt(5.0)) + (sqrt(5.0) + 1.0)))));
}

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. Simplified0.5

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

  4. Applied flip3--_binary640.5

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

  5. Simplified0.5

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

  7. Applied flip--_binary640.6

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

  8. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\]
  9. Using strategy rm

  10. Applied flip3--_binary640.9

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + 1.5 \cdot \left(\cos x \cdot \color{blue}{\frac{{\left(\sqrt{5}\right)}^{3} - {1}^{3}}{\sqrt{5} \cdot \sqrt{5} + \left(1 \cdot 1 + \sqrt{5} \cdot 1\right)}}\right)\right)}\]

  11. Applied associate-*r/_binary640.8

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + 1.5 \cdot \color{blue}{\frac{\cos x \cdot \left({\left(\sqrt{5}\right)}^{3} - {1}^{3}\right)}{\sqrt{5} \cdot \sqrt{5} + \left(1 \cdot 1 + \sqrt{5} \cdot 1\right)}}\right)}\]

  12. Applied associate-*r/_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \color{blue}{\frac{1.5 \cdot \left(\cos x \cdot \left({\left(\sqrt{5}\right)}^{3} - {1}^{3}\right)\right)}{\sqrt{5} \cdot \sqrt{5} + \left(1 \cdot 1 + \sqrt{5} \cdot 1\right)}}\right)}\]

  13. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{\color{blue}{1.5 \cdot \left(\cos x \cdot \left(-1 + 5 \cdot \sqrt{5}\right)\right)}}{\sqrt{5} \cdot \sqrt{5} + \left(1 \cdot 1 + \sqrt{5} \cdot 1\right)}\right)}\]

  14. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{{\cos x}^{3} - {\cos y}^{3}}{\cos x \cdot \cos x + \cos y \cdot \left(\cos x + \cos y\right)}}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{3 + \sqrt{5}}\right) + \frac{1.5 \cdot \left(\cos x \cdot \left(-1 + 5 \cdot \sqrt{5}\right)\right)}{\sqrt{5} \cdot \sqrt{5} + \left(\sqrt{5} + 1\right)}\right)}\]

Reproduce

herbie shell --seed 2020260 
(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))))))