Average Error: 0.5 → 0.4
Time: 3.2min
Precision: binary64
Cost: 3712
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \sqrt{5}\right)}}\]
\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(\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 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \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
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (+
   3.0
   (/
    (+
     (* (* (cos y) 6.0) (+ 1.0 (sqrt 5.0)))
     (* (* (cos x) 6.0) (+ 3.0 (sqrt 5.0))))
    (* (+ 1.0 (sqrt 5.0)) (+ 3.0 (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 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((((cos(y) * 6.0) * (1.0 + sqrt(5.0))) + ((cos(x) * 6.0) * (3.0 + sqrt(5.0)))) / ((1.0 + sqrt(5.0)) * (3.0 + 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
Alternative 1
Accuracy0.4
Cost4480
\[\frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \frac{\sqrt[3]{\sin y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \sqrt{5}\right)}}\]
Alternative 2
Accuracy31.8
Cost4672
\[\frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \left(\frac{\sqrt{\sin y}}{\sqrt[3]{16} \cdot \sqrt[3]{16}} \cdot \frac{\sqrt{\sin y}}{\sqrt[3]{16}}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \sqrt{5}\right)}}\]
Alternative 3
Accuracy31.8
Cost4352
\[\frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \left(\frac{\sqrt{\sin y}}{4} \cdot \frac{\sqrt{\sin y}}{4}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \sqrt{5}\right)}}\]

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 flip--_binary64_48270.5

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

  5. Simplified0.4

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

  7. Applied flip--_binary64_48270.9

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

  8. Applied associate-*r/_binary64_47940.8

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

  9. Applied associate-*r/_binary64_47940.5

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

  10. Applied associate-*r/_binary64_47940.5

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

  11. Applied associate-*r/_binary64_47940.5

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

  12. Applied frac-add_binary64_48600.8

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  16. Applied sub-neg_binary64_48450.4

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

  18. Applied *-un-lft-identity_binary64_48520.4

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

  19. Applied sqrt-prod_binary64_48680.4

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

  20. Applied associate-*l*_binary64_47930.4

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

  21. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\left(\cos y \cdot 6\right) \cdot \left(1 + \sqrt{5}\right) + \left(\cos x \cdot 6\right) \cdot \left(3 + \sqrt{5}\right)}{\left(1 + \sqrt{5}\right) \cdot \left(3 + \sqrt{5}\right)}}\]

Reproduce

herbie shell --seed 2020322 
(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))))))