Average Error: 0.4 → 0.4
Time: 14.6s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}\right)\right) \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sin x + \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y \cdot 1\right) + \mathsf{fma}\left(-\cos y, 1, \cos y\right)\right), 2\right)}{3}}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}\]
\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{\frac{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}\right)\right) \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sin x + \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y \cdot 1\right) + \mathsf{fma}\left(-\cos y, 1, \cos y\right)\right), 2\right)}{3}}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}
double f(double x, double y) {
        double r175501 = 2.0;
        double r175502 = sqrt(r175501);
        double r175503 = x;
        double r175504 = sin(r175503);
        double r175505 = y;
        double r175506 = sin(r175505);
        double r175507 = 16.0;
        double r175508 = r175506 / r175507;
        double r175509 = r175504 - r175508;
        double r175510 = r175502 * r175509;
        double r175511 = r175504 / r175507;
        double r175512 = r175506 - r175511;
        double r175513 = r175510 * r175512;
        double r175514 = cos(r175503);
        double r175515 = cos(r175505);
        double r175516 = r175514 - r175515;
        double r175517 = r175513 * r175516;
        double r175518 = r175501 + r175517;
        double r175519 = 3.0;
        double r175520 = 1.0;
        double r175521 = 5.0;
        double r175522 = sqrt(r175521);
        double r175523 = r175522 - r175520;
        double r175524 = r175523 / r175501;
        double r175525 = r175524 * r175514;
        double r175526 = r175520 + r175525;
        double r175527 = r175519 - r175522;
        double r175528 = r175527 / r175501;
        double r175529 = r175528 * r175515;
        double r175530 = r175526 + r175529;
        double r175531 = r175519 * r175530;
        double r175532 = r175518 / r175531;
        return r175532;
}
↓
double f(double x, double y) {
        double r175533 = 2.0;
        double r175534 = sqrt(r175533);
        double r175535 = x;
        double r175536 = sin(r175535);
        double r175537 = r175536 * r175536;
        double r175538 = y;
        double r175539 = sin(r175538);
        double r175540 = 16.0;
        double r175541 = r175539 / r175540;
        double r175542 = r175541 * r175541;
        double r175543 = r175537 - r175542;
        double r175544 = cbrt(r175543);
        double r175545 = r175544 * r175544;
        double r175546 = r175534 * r175545;
        double r175547 = r175546 * r175544;
        double r175548 = r175536 + r175541;
        double r175549 = r175547 / r175548;
        double r175550 = r175536 / r175540;
        double r175551 = r175539 - r175550;
        double r175552 = cos(r175535);
        double r175553 = cbrt(r175552);
        double r175554 = r175553 * r175553;
        double r175555 = cos(r175538);
        double r175556 = 1.0;
        double r175557 = r175555 * r175556;
        double r175558 = -r175557;
        double r175559 = fma(r175554, r175553, r175558);
        double r175560 = -r175555;
        double r175561 = fma(r175560, r175556, r175555);
        double r175562 = r175559 + r175561;
        double r175563 = r175551 * r175562;
        double r175564 = fma(r175549, r175563, r175533);
        double r175565 = 3.0;
        double r175566 = r175564 / r175565;
        double r175567 = 5.0;
        double r175568 = sqrt(r175567);
        double r175569 = r175565 - r175568;
        double r175570 = r175569 / r175533;
        double r175571 = 1.0;
        double r175572 = r175568 - r175571;
        double r175573 = r175572 / r175533;
        double r175574 = fma(r175573, r175552, r175571);
        double r175575 = fma(r175570, r175555, r175574);
        double r175576 = r175566 / r175575;
        return r175576;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied prod-diff0.5

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

  7. Applied distribute-lft-in0.5

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  10. Simplified0.4

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

  12. Applied flip--0.4

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

  13. Applied associate-*r/0.4

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

  15. Applied add-cube-cbrt0.4

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

  16. Applied associate-*r*0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))