Average Error: 0.5 → 0.4
Time: 15.2s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\sqrt{2} \cdot \sqrt[3]{{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}^{3}}\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{6 \cdot \left(\cos y \cdot \left(\sqrt{5} + 1\right) + \cos x \cdot \left(3 + \sqrt{5}\right)\right)}{\left(\sqrt{5} + 1\right) \cdot \left(3 + \sqrt{5}\right)}}\]
\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(\sqrt{2} \cdot \sqrt[3]{{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}^{3}}\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{6 \cdot \left(\cos y \cdot \left(\sqrt{5} + 1\right) + \cos x \cdot \left(3 + \sqrt{5}\right)\right)}{\left(\sqrt{5} + 1\right) \cdot \left(3 + \sqrt{5}\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)
     (cbrt
      (pow (* (- (sin y) (/ (sin x) 16.0)) (- (sin x) (/ (sin y) 16.0))) 3.0)))
    (- (cos x) (cos y))))
  (+
   3.0
   (/
    (* 6.0 (+ (* (cos y) (+ (sqrt 5.0) 1.0)) (* (cos x) (+ 3.0 (sqrt 5.0)))))
    (* (+ (sqrt 5.0) 1.0) (+ 3.0 (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) * cbrt(pow(((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0))), 3.0))) * (cos(x) - cos(y)))) / (3.0 + ((6.0 * ((cos(y) * (sqrt(5.0) + 1.0)) + (cos(x) * (3.0 + sqrt(5.0))))) / ((sqrt(5.0) + 1.0) * (3.0 + 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. Simplified0.5

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

  4. Applied flip--_binary64_38100.7

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

  5. Applied associate-*r/_binary64_38450.6

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

  6. Applied associate-*r/_binary64_38450.6

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

  7. Simplified0.4

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

  9. Applied flip--_binary64_38100.5

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

  10. Applied associate-*r/_binary64_38450.5

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

  11. Applied associate-*r/_binary64_38450.5

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

  12. Applied frac-add_binary64_37740.5

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

  13. Simplified0.4

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

  15. Applied associate-*l*_binary64_38440.4

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

  17. Applied add-cbrt-cube_binary64_37560.4

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

  18. Simplified0.4

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

  19. Final simplification0.4

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \sqrt[3]{{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}^{3}}\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{6 \cdot \left(\cos y \cdot \left(\sqrt{5} + 1\right) + \cos x \cdot \left(3 + \sqrt{5}\right)\right)}{\left(\sqrt{5} + 1\right) \cdot \left(3 + \sqrt{5}\right)}}\]

Reproduce

herbie shell --seed 2020231 
(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))))))