Average Error: 0.5 → 0.5
Time: 39.6s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{1}{3}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, \mathsf{fma}\left(\frac{\cos y}{2}, 3 - \sqrt{5}, 1\right)\right)} \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), \sin x - \frac{\sin y}{16}, 2\right)\]
\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{1}{3}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, \mathsf{fma}\left(\frac{\cos y}{2}, 3 - \sqrt{5}, 1\right)\right)} \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right), \sin x - \frac{\sin y}{16}, 2\right)
double f(double x, double y) {
        double r6882343 = 2.0;
        double r6882344 = sqrt(r6882343);
        double r6882345 = x;
        double r6882346 = sin(r6882345);
        double r6882347 = y;
        double r6882348 = sin(r6882347);
        double r6882349 = 16.0;
        double r6882350 = r6882348 / r6882349;
        double r6882351 = r6882346 - r6882350;
        double r6882352 = r6882344 * r6882351;
        double r6882353 = r6882346 / r6882349;
        double r6882354 = r6882348 - r6882353;
        double r6882355 = r6882352 * r6882354;
        double r6882356 = cos(r6882345);
        double r6882357 = cos(r6882347);
        double r6882358 = r6882356 - r6882357;
        double r6882359 = r6882355 * r6882358;
        double r6882360 = r6882343 + r6882359;
        double r6882361 = 3.0;
        double r6882362 = 1.0;
        double r6882363 = 5.0;
        double r6882364 = sqrt(r6882363);
        double r6882365 = r6882364 - r6882362;
        double r6882366 = r6882365 / r6882343;
        double r6882367 = r6882366 * r6882356;
        double r6882368 = r6882362 + r6882367;
        double r6882369 = r6882361 - r6882364;
        double r6882370 = r6882369 / r6882343;
        double r6882371 = r6882370 * r6882357;
        double r6882372 = r6882368 + r6882371;
        double r6882373 = r6882361 * r6882372;
        double r6882374 = r6882360 / r6882373;
        return r6882374;
}
↓
double f(double x, double y) {
        double r6882375 = 1.0;
        double r6882376 = 3.0;
        double r6882377 = r6882375 / r6882376;
        double r6882378 = x;
        double r6882379 = cos(r6882378);
        double r6882380 = 5.0;
        double r6882381 = sqrt(r6882380);
        double r6882382 = 1.0;
        double r6882383 = r6882381 - r6882382;
        double r6882384 = 2.0;
        double r6882385 = r6882383 / r6882384;
        double r6882386 = y;
        double r6882387 = cos(r6882386);
        double r6882388 = r6882387 / r6882384;
        double r6882389 = r6882376 - r6882381;
        double r6882390 = fma(r6882388, r6882389, r6882382);
        double r6882391 = fma(r6882379, r6882385, r6882390);
        double r6882392 = r6882377 / r6882391;
        double r6882393 = sqrt(r6882384);
        double r6882394 = r6882379 - r6882387;
        double r6882395 = sin(r6882386);
        double r6882396 = sin(r6882378);
        double r6882397 = 16.0;
        double r6882398 = r6882396 / r6882397;
        double r6882399 = r6882395 - r6882398;
        double r6882400 = r6882394 * r6882399;
        double r6882401 = r6882393 * r6882400;
        double r6882402 = r6882395 / r6882397;
        double r6882403 = r6882396 - r6882402;
        double r6882404 = fma(r6882401, r6882403, r6882384);
        double r6882405 = r6882392 * r6882404;
        return r6882405;
}

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

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

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

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

  5. Applied div-inv0.5

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019192 +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))))))