Average Error: 0.5 → 0.5
Time: 16.1s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \frac{\sin y}{{\left(\sqrt[3]{16}\right)}^{3}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
\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{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \frac{\sin y}{{\left(\sqrt[3]{16}\right)}^{3}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{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
   (*
    (*
     (-
      (* (sqrt 2.0) (sin x))
      (* (sqrt 2.0) (/ (sin y) (pow (cbrt 16.0) 3.0))))
     (- (sin y) (/ (sin x) 16.0)))
    (cbrt (pow (- (cos x) (cos y)) 3.0))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (- (sqrt 5.0) 1.0) 2.0)))
    (* (cos y) (/ (- 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 + ((((sqrt(2.0) * sin(x)) - (sqrt(2.0) * (sin(y) / pow(cbrt(16.0), 3.0)))) * (sin(y) - (sin(x) / 16.0))) * cbrt(pow((cos(x) - cos(y)), 3.0)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) - 1.0) / 2.0))) + (cos(y) * ((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. Applied add-cube-cbrt_binary640.5

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

  3. Applied add-cube-cbrt_binary640.5

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

  4. Applied times-frac_binary640.5

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

  5. Applied cancel-sign-sub-inv_binary640.5

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

  6. Applied distribute-rgt-in_binary640.5

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

  7. Simplified0.5

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

  8. Simplified0.5

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

  9. Applied add-cbrt-cube_binary640.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2022104 
(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))))))