Average Error: 0.5 → 0.4
Time: 10.8s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \frac{4}{\sqrt{5} + 3}\right)\right)}\]
\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(\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)}{\cos x \cdot \left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \frac{4}{\sqrt{5} + 3}\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
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (+
   (*
    (cos x)
    (*
     (/ 1.0 (sqrt 0.6666666666666666))
     (/ (- (sqrt 5.0) 1.0) (sqrt 0.6666666666666666))))
   (+ 3.0 (* 1.5 (* (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0))))))))
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) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / ((cos(x) * ((1.0 / sqrt(0.6666666666666666)) * ((sqrt(5.0) - 1.0) / sqrt(0.6666666666666666)))) + (3.0 + (1.5 * (cos(y) * (4.0 / (sqrt(5.0) + 3.0))))));
}

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

    \[\leadsto \color{blue}{\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{\cos x \cdot \left(\sqrt{5} - 1\right)}{0.6666666666666666} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}}\]
  3. Using strategy rm

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

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{\cos x \cdot \left(\sqrt{5} - 1\right)}{\color{blue}{1 \cdot 0.6666666666666666}} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]

  5. Applied times-frac_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{\cos x}{1} \cdot \frac{\sqrt{5} - 1}{0.6666666666666666}} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]

  6. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\cos x} \cdot \frac{\sqrt{5} - 1}{0.6666666666666666} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \frac{\sqrt{5} - 1}{\color{blue}{\sqrt{0.6666666666666666} \cdot \sqrt{0.6666666666666666}}} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]

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

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \frac{\color{blue}{1 \cdot \left(\sqrt{5} - 1\right)}}{\sqrt{0.6666666666666666} \cdot \sqrt{0.6666666666666666}} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]

  10. Applied times-frac_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \color{blue}{\left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right)} + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\]
  11. Using strategy rm

  12. Applied flip--_binary640.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right)\right)}\]

  13. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)\right)}\]

  14. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\cos x \cdot \left(\frac{1}{\sqrt{0.6666666666666666}} \cdot \frac{\sqrt{5} - 1}{\sqrt{0.6666666666666666}}\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \frac{4}{\sqrt{5} + 3}\right)\right)}\]

Reproduce

herbie shell --seed 2020220 
(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))))))