Average Error: 0.5 → 0.4
Time: 18.2s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{\sin y}}\right)}^{6}}{{\left(\sqrt{\sqrt[3]{16}}\right)}^{3}} \cdot \frac{\sqrt[3]{\sin y}}{{\left(\left|\sqrt{\sqrt[3]{\sqrt[3]{16}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{16}}}\right|\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{\sqrt[3]{16}}}\right)}^{3}}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\]
\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 \left(\frac{{\left(\sqrt[3]{\sqrt[3]{\sin y}}\right)}^{6}}{{\left(\sqrt{\sqrt[3]{16}}\right)}^{3}} \cdot \frac{\sqrt[3]{\sin y}}{{\left(\left|\sqrt{\sqrt[3]{\sqrt[3]{16}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{16}}}\right|\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{\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 + \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)}
(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)
       (*
        (/ (pow (cbrt (cbrt (sin y))) 6.0) (pow (sqrt (cbrt 16.0)) 3.0))
        (/
         (cbrt (sin y))
         (*
          (pow
           (fabs (* (sqrt (cbrt (cbrt 16.0))) (sqrt (cbrt (cbrt 16.0)))))
           3.0)
          (pow (sqrt (cbrt (cbrt 16.0))) 3.0))))))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (+
   3.0
   (+
    (* (cos y) (* 1.5 (- 3.0 (sqrt 5.0))))
    (* 1.5 (* (cos x) (- (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(x)) - (sqrt(2.0) * ((pow(cbrt(cbrt(sin(y))), 6.0) / pow(sqrt(cbrt(16.0)), 3.0)) * (cbrt(sin(y)) / (pow(fabs(sqrt(cbrt(cbrt(16.0))) * sqrt(cbrt(cbrt(16.0)))), 3.0) * pow(sqrt(cbrt(cbrt(16.0))), 3.0)))))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((cos(y) * (1.5 * (3.0 - sqrt(5.0)))) + (1.5 * (cos(x) * (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. Simplified0.5

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

  4. Applied add-cube-cbrt_binary64_45460.4

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

  5. Applied add-cube-cbrt_binary64_45460.4

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

  6. Applied times-frac_binary64_45170.4

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

  7. Applied cancel-sign-sub-inv_binary64_44770.4

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

  8. Applied distribute-rgt-in_binary64_44610.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-sqr-sqrt_binary64_45330.4

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

  13. Applied add-cube-cbrt_binary64_45460.4

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

  14. Applied times-frac_binary64_45170.4

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

  15. Applied unpow-prod-down_binary64_45900.4

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

  16. Simplified0.4

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

  17. Simplified0.4

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

  19. Applied add-cube-cbrt_binary64_45460.4

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

  20. Applied sqrt-prod_binary64_45270.4

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

  21. Applied unpow-prod-down_binary64_45900.4

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

  22. Simplified0.4

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

  24. Applied add-sqr-sqrt_binary64_45330.4

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

  25. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \sin x - \sqrt{2} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{\sin y}}\right)}^{6}}{{\left(\sqrt{\sqrt[3]{16}}\right)}^{3}} \cdot \frac{\sqrt[3]{\sin y}}{{\left(\left|\sqrt{\sqrt[3]{\sqrt[3]{16}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{16}}}\right|\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{\sqrt[3]{16}}}\right)}^{3}}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\]

Reproduce

herbie shell --seed 2021019 
(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))))))