Average Error: 0.5 → 0.4
Time: 18.3s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\log \left(e^{\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \left(\cos x \cdot -4\right)}{-1 - \sqrt{5}}\right)}\]
\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(\log \left(e^{\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{1.5 \cdot \left(\cos x \cdot -4\right)}{-1 - \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
   (*
    (*
     (log (exp (* (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 (- 3.0 (sqrt 5.0))))
    (/ (* 1.5 (* (cos x) -4.0)) (- -1.0 (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 + ((log(exp(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 * (3.0 - sqrt(5.0)))) + ((1.5 * (cos(x) * -4.0)) / (-1.0 - 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. Simplified0.4

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

  4. Applied add-sqr-sqrt_binary64_41920.4

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

  5. Applied associate-*l*_binary64_41110.4

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

  6. Simplified0.4

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

  8. Applied add-log-exp_binary64_42090.4

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

  10. Applied flip-+_binary64_41440.4

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

  11. Applied associate-*r/_binary64_41120.4

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

  12. Applied associate-*l/_binary64_41130.4

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

  13. Applied associate-*r/_binary64_41120.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020342 
(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))))))