Average Error: 0.5 → 0.4
Time: 36.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\left(\sin x + \left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right)\right) \cdot \sqrt{2} + \left(\left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) + {\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) \cdot \sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}\]
\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(\left(\sin x + \left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right)\right) \cdot \sqrt{2} + \left(\left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) + {\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) \cdot \sqrt{2}, \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 r124404 = 2.0;
        double r124405 = sqrt(r124404);
        double r124406 = x;
        double r124407 = sin(r124406);
        double r124408 = y;
        double r124409 = sin(r124408);
        double r124410 = 16.0;
        double r124411 = r124409 / r124410;
        double r124412 = r124407 - r124411;
        double r124413 = r124405 * r124412;
        double r124414 = r124407 / r124410;
        double r124415 = r124409 - r124414;
        double r124416 = r124413 * r124415;
        double r124417 = cos(r124406);
        double r124418 = cos(r124408);
        double r124419 = r124417 - r124418;
        double r124420 = r124416 * r124419;
        double r124421 = r124404 + r124420;
        double r124422 = 3.0;
        double r124423 = 1.0;
        double r124424 = 5.0;
        double r124425 = sqrt(r124424);
        double r124426 = r124425 - r124423;
        double r124427 = r124426 / r124404;
        double r124428 = r124427 * r124417;
        double r124429 = r124423 + r124428;
        double r124430 = r124422 - r124425;
        double r124431 = r124430 / r124404;
        double r124432 = r124431 * r124418;
        double r124433 = r124429 + r124432;
        double r124434 = r124422 * r124433;
        double r124435 = r124421 / r124434;
        return r124435;
}
↓
double f(double x, double y) {
        double r124436 = x;
        double r124437 = sin(r124436);
        double r124438 = y;
        double r124439 = sin(r124438);
        double r124440 = cbrt(r124439);
        double r124441 = 16.0;
        double r124442 = cbrt(r124441);
        double r124443 = r124440 / r124442;
        double r124444 = 3.0;
        double r124445 = pow(r124443, r124444);
        double r124446 = -r124445;
        double r124447 = r124437 + r124446;
        double r124448 = 2.0;
        double r124449 = sqrt(r124448);
        double r124450 = r124447 * r124449;
        double r124451 = r124446 + r124445;
        double r124452 = r124451 * r124449;
        double r124453 = r124450 + r124452;
        double r124454 = r124437 / r124441;
        double r124455 = r124439 - r124454;
        double r124456 = cos(r124436);
        double r124457 = cos(r124438);
        double r124458 = r124456 - r124457;
        double r124459 = r124455 * r124458;
        double r124460 = fma(r124453, r124459, r124448);
        double r124461 = 3.0;
        double r124462 = r124460 / r124461;
        double r124463 = 5.0;
        double r124464 = sqrt(r124463);
        double r124465 = r124461 - r124464;
        double r124466 = r124465 / r124448;
        double r124467 = 1.0;
        double r124468 = r124464 - r124467;
        double r124469 = r124468 / r124448;
        double r124470 = fma(r124456, r124469, r124467);
        double r124471 = fma(r124457, r124466, r124470);
        double r124472 = r124462 / r124471;
        return r124472;
}

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 add-cube-cbrt0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied add-sqr-sqrt32.2

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

  8. Applied prod-diff32.2

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

  9. Applied distribute-lft-in32.2

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  (/ (+ 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))))))