Average Error: 0.5 → 0.4
Time: 4.7min
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + 3 \cdot \frac{\left(3 + \sqrt{5}\right) \cdot \left(\frac{\cos x}{2} \cdot \left(5 + -1\right)\right) + \left(\left(3 \cdot 3 - 5\right) \cdot \frac{\cos y}{2}\right) \cdot \left(\sqrt{5} - -1\right)}{\left(3 + \sqrt{5}\right) \cdot \left(\sqrt{5} - -1\right)}}\]
\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 + \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + 3 \cdot \frac{\left(3 + \sqrt{5}\right) \cdot \left(\frac{\cos x}{2} \cdot \left(5 + -1\right)\right) + \left(\left(3 \cdot 3 - 5\right) \cdot \frac{\cos y}{2}\right) \cdot \left(\sqrt{5} - -1\right)}{\left(3 + \sqrt{5}\right) \cdot \left(\sqrt{5} - -1\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 y) (/ (sin x) 16.0))
     (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos y))))))
  (+
   3.0
   (*
    3.0
    (/
     (+
      (* (+ 3.0 (sqrt 5.0)) (* (/ (cos x) 2.0) (+ 5.0 -1.0)))
      (* (* (- (* 3.0 3.0) 5.0) (/ (cos y) 2.0)) (- (sqrt 5.0) -1.0)))
     (* (+ 3.0 (sqrt 5.0)) (- (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 + (sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(y)))))) / (3.0 + (3.0 * ((((3.0 + sqrt(5.0)) * ((cos(x) / 2.0) * (5.0 + -1.0))) + ((((3.0 * 3.0) - 5.0) * (cos(y) / 2.0)) * (sqrt(5.0) - -1.0))) / ((3.0 + sqrt(5.0)) * (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

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

  4. Applied flip--_binary64_61910.5

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

  5. Applied associate-*r/_binary64_61580.5

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

  6. Applied flip-+_binary64_61900.7

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

  7. Applied associate-*r/_binary64_61580.6

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

  8. Applied frac-add_binary64_62240.6

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021176 
(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))))))