Average Error: 0.5 → 0.5
Time: 13.3s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\log \left(e^{3 - \sqrt{5}}\right)}{2}\right)}\]
\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{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\log \left(e^{3 - \sqrt{5}}\right)}{2}\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
    (*
     (- (cos x) (cos y))
     (*
      (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
      (- (sin y) (/ (sin x) 16.0)))))
   3.0)
  (+
   (* (cos x) (- (* 0.5 (sqrt 5.0)) 0.5))
   (+ 1.0 (* (cos y) (/ (log (exp (- 3.0 (sqrt 5.0)))) 2.0))))))
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 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / 3.0) / ((cos(x) * ((0.5 * sqrt(5.0)) - 0.5)) + (1.0 + (cos(y) * (log(exp(3.0 - 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 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. Using strategy rm

  3. Applied associate-/r*_binary640.5

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

  4. Simplified0.5

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

  6. Applied *-un-lft-identity_binary640.5

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

  7. Applied add-sqr-sqrt_binary640.8

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

  8. Applied difference-of-sqr-1_binary640.5

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

  9. Applied times-frac_binary640.5

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

  10. Applied associate-*l*_binary640.5

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

  11. Simplified0.5

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

  12. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}}\]
  13. Using strategy rm

  14. Applied add-log-exp_binary640.5

    \[\leadsto \frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{3 - \color{blue}{\log \left(e^{\sqrt{5}}\right)}}{2}\right)}\]

  15. Applied add-log-exp_binary640.5

    \[\leadsto \frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\color{blue}{\log \left(e^{3}\right)} - \log \left(e^{\sqrt{5}}\right)}{2}\right)}\]

  16. Applied diff-log_binary640.5

    \[\leadsto \frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\color{blue}{\log \left(\frac{e^{3}}{e^{\sqrt{5}}}\right)}}{2}\right)}\]

  17. Simplified0.5

    \[\leadsto \frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\log \color{blue}{\left(e^{3 - \sqrt{5}}\right)}}{2}\right)}\]

  18. Final simplification0.5

    \[\leadsto \frac{\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3}}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(1 + \cos y \cdot \frac{\log \left(e^{3 - \sqrt{5}}\right)}{2}\right)}\]

Reproduce

herbie shell --seed 2020274 
(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))))))