Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5

Average Error: 0.5 → 0.5

Time: 12.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r198527 = 2.0;
        double r198528 = sqrt(r198527);
        double r198529 = x;
        double r198530 = sin(r198529);
        double r198531 = y;
        double r198532 = sin(r198531);
        double r198533 = 16.0;
        double r198534 = r198532 / r198533;
        double r198535 = r198530 - r198534;
        double r198536 = r198528 * r198535;
        double r198537 = r198530 / r198533;
        double r198538 = r198532 - r198537;
        double r198539 = r198536 * r198538;
        double r198540 = cos(r198529);
        double r198541 = cos(r198531);
        double r198542 = r198540 - r198541;
        double r198543 = r198539 * r198542;
        double r198544 = r198527 + r198543;
        double r198545 = 3.0;
        double r198546 = 1.0;
        double r198547 = 5.0;
        double r198548 = sqrt(r198547);
        double r198549 = r198548 - r198546;
        double r198550 = r198549 / r198527;
        double r198551 = r198550 * r198540;
        double r198552 = r198546 + r198551;
        double r198553 = r198545 - r198548;
        double r198554 = r198553 / r198527;
        double r198555 = r198554 * r198541;
        double r198556 = r198552 + r198555;
        double r198557 = r198545 * r198556;
        double r198558 = r198544 / r198557;
        return r198558;
}

↓

double f(double x, double y) {
        double r198559 = 2.0;
        double r198560 = sqrt(r198559);
        double r198561 = x;
        double r198562 = sin(r198561);
        double r198563 = cbrt(r198562);
        double r198564 = r198563 * r198563;
        double r198565 = y;
        double r198566 = sin(r198565);
        double r198567 = cbrt(r198566);
        double r198568 = 16.0;
        double r198569 = cbrt(r198568);
        double r198570 = r198567 / r198569;
        double r198571 = r198567 * r198567;
        double r198572 = r198569 * r198569;
        double r198573 = r198571 / r198572;
        double r198574 = r198570 * r198573;
        double r198575 = -r198574;
        double r198576 = fma(r198564, r198563, r198575);
        double r198577 = r198560 * r198576;
        double r198578 = cbrt(r198572);
        double r198579 = cbrt(r198569);
        double r198580 = r198578 * r198579;
        double r198581 = r198567 / r198580;
        double r198582 = -r198581;
        double r198583 = r198582 + r198570;
        double r198584 = r198573 * r198583;
        double r198585 = r198584 * r198560;
        double r198586 = r198577 + r198585;
        double r198587 = r198562 / r198568;
        double r198588 = r198566 - r198587;
        double r198589 = r198586 * r198588;
        double r198590 = cos(r198561);
        double r198591 = cos(r198565);
        double r198592 = r198590 - r198591;
        double r198593 = r198589 * r198592;
        double r198594 = r198559 + r198593;
        double r198595 = 3.0;
        double r198596 = 1.0;
        double r198597 = 5.0;
        double r198598 = sqrt(r198597);
        double r198599 = r198598 - r198596;
        double r198600 = r198599 / r198559;
        double r198601 = r198600 * r198590;
        double r198602 = r198596 + r198601;
        double r198603 = r198595 - r198598;
        double r198604 = r198603 / r198559;
        double r198605 = r198604 * r198591;
        double r198606 = r198602 + r198605;
        double r198607 = r198595 * r198606;
        double r198608 = r198594 / r198607;
        return r198608;
}

Error

Bits error versus x

Bits error versus y

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. Using strategy rm

3. Applied add-cube-cbrt 0.4

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

4. Applied add-cube-cbrt 0.5

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

5. Applied times-frac 0.5

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

6. Applied add-cube-cbrt 0.5

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

7. Applied prod-diff 0.5

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

8. Applied distribute-lft-in 0.5

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

9. Simplified 0.5

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

11. Applied add-cube-cbrt 0.5

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

12. Applied cbrt-prod 0.5

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

13. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019362 +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))))))