Average Error: 0.5 → 0.5
Time: 37.2s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \sqrt{2} \cdot \left(\left(-\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}\]
\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(\sqrt{2} \cdot \mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \sqrt{2} \cdot \left(\left(-\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}
double f(double x, double y) {
        double r198420 = 2.0;
        double r198421 = sqrt(r198420);
        double r198422 = x;
        double r198423 = sin(r198422);
        double r198424 = y;
        double r198425 = sin(r198424);
        double r198426 = 16.0;
        double r198427 = r198425 / r198426;
        double r198428 = r198423 - r198427;
        double r198429 = r198421 * r198428;
        double r198430 = r198423 / r198426;
        double r198431 = r198425 - r198430;
        double r198432 = r198429 * r198431;
        double r198433 = cos(r198422);
        double r198434 = cos(r198424);
        double r198435 = r198433 - r198434;
        double r198436 = r198432 * r198435;
        double r198437 = r198420 + r198436;
        double r198438 = 3.0;
        double r198439 = 1.0;
        double r198440 = 5.0;
        double r198441 = sqrt(r198440);
        double r198442 = r198441 - r198439;
        double r198443 = r198442 / r198420;
        double r198444 = r198443 * r198433;
        double r198445 = r198439 + r198444;
        double r198446 = r198438 - r198441;
        double r198447 = r198446 / r198420;
        double r198448 = r198447 * r198434;
        double r198449 = r198445 + r198448;
        double r198450 = r198438 * r198449;
        double r198451 = r198437 / r198450;
        return r198451;
}
↓
double f(double x, double y) {
        double r198452 = 2.0;
        double r198453 = sqrt(r198452);
        double r198454 = x;
        double r198455 = sin(r198454);
        double r198456 = cbrt(r198455);
        double r198457 = r198456 * r198456;
        double r198458 = y;
        double r198459 = sin(r198458);
        double r198460 = cbrt(r198459);
        double r198461 = 3.0;
        double r198462 = pow(r198460, r198461);
        double r198463 = 16.0;
        double r198464 = r198462 / r198463;
        double r198465 = -r198464;
        double r198466 = fma(r198457, r198456, r198465);
        double r198467 = r198453 * r198466;
        double r198468 = r198465 + r198464;
        double r198469 = r198453 * r198468;
        double r198470 = r198467 + r198469;
        double r198471 = r198455 / r198463;
        double r198472 = r198459 - r198471;
        double r198473 = cos(r198454);
        double r198474 = cos(r198458);
        double r198475 = r198473 - r198474;
        double r198476 = r198472 * r198475;
        double r198477 = fma(r198470, r198476, r198452);
        double r198478 = 3.0;
        double r198479 = r198477 / r198478;
        double r198480 = 5.0;
        double r198481 = sqrt(r198480);
        double r198482 = r198478 - r198481;
        double r198483 = r198482 / r198452;
        double r198484 = 1.0;
        double r198485 = r198481 - r198484;
        double r198486 = r198485 / r198452;
        double r198487 = fma(r198473, r198486, r198484);
        double r198488 = fma(r198474, r198483, r198487);
        double r198489 = r198479 / r198488;
        return r198489;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied prod-diff0.5

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

  9. Applied distribute-lft-in0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +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))))))