Average Error: 0.5 → 0.4
Time: 19.3s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sin x \cdot \sqrt{2} + \sqrt{2} \cdot \left(\frac{-1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \frac{\sin y}{\sqrt[3]{16}}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \left(\cos x \cdot 1.5\right) \cdot \left(\sqrt{5} - 1\right)\right)}\]
\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(\sin x \cdot \sqrt{2} + \sqrt{2} \cdot \left(\frac{-1}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \frac{\sin y}{\sqrt[3]{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(\cos x \cdot 1.5\right) \cdot \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
   (*
    (*
     (+
      (* (sin x) (sqrt 2.0))
      (*
       (sqrt 2.0)
       (* (/ -1.0 (* (cbrt 16.0) (cbrt 16.0))) (/ (sin y) (cbrt 16.0)))))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (+
   3.0
   (+
    (* (cos y) (* 1.5 (- 3.0 (sqrt 5.0))))
    (* (* (cos x) 1.5) (- (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 + ((((sin(x) * sqrt(2.0)) + (sqrt(2.0) * ((-1.0 / (cbrt(16.0) * cbrt(16.0))) * (sin(y) / cbrt(16.0))))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((cos(y) * (1.5 * (3.0 - sqrt(5.0)))) + ((cos(x) * 1.5) * (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. Simplified0.4

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

  4. Applied associate-*r*_binary640.4

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt_binary640.4

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

  8. Applied *-un-lft-identity_binary640.4

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

  9. Applied times-frac_binary640.4

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

  10. Applied cancel-sign-sub-inv_binary640.4

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

  11. Applied distribute-rgt-in_binary640.4

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

  12. Final simplification0.4

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))))