Average Error: 0.5 → 0.5
Time: 45.0s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\log \left(e^{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right)}\right) + 2}{\left(\cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right)\right) \cdot 3}\]
\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{\log \left(e^{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right)}\right) + 2}{\left(\cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right)\right) \cdot 3}
double f(double x, double y) {
        double r8907465 = 2.0;
        double r8907466 = sqrt(r8907465);
        double r8907467 = x;
        double r8907468 = sin(r8907467);
        double r8907469 = y;
        double r8907470 = sin(r8907469);
        double r8907471 = 16.0;
        double r8907472 = r8907470 / r8907471;
        double r8907473 = r8907468 - r8907472;
        double r8907474 = r8907466 * r8907473;
        double r8907475 = r8907468 / r8907471;
        double r8907476 = r8907470 - r8907475;
        double r8907477 = r8907474 * r8907476;
        double r8907478 = cos(r8907467);
        double r8907479 = cos(r8907469);
        double r8907480 = r8907478 - r8907479;
        double r8907481 = r8907477 * r8907480;
        double r8907482 = r8907465 + r8907481;
        double r8907483 = 3.0;
        double r8907484 = 1.0;
        double r8907485 = 5.0;
        double r8907486 = sqrt(r8907485);
        double r8907487 = r8907486 - r8907484;
        double r8907488 = r8907487 / r8907465;
        double r8907489 = r8907488 * r8907478;
        double r8907490 = r8907484 + r8907489;
        double r8907491 = r8907483 - r8907486;
        double r8907492 = r8907491 / r8907465;
        double r8907493 = r8907492 * r8907479;
        double r8907494 = r8907490 + r8907493;
        double r8907495 = r8907483 * r8907494;
        double r8907496 = r8907482 / r8907495;
        return r8907496;
}
↓
double f(double x, double y) {
        double r8907497 = x;
        double r8907498 = cos(r8907497);
        double r8907499 = y;
        double r8907500 = cos(r8907499);
        double r8907501 = r8907498 - r8907500;
        double r8907502 = sin(r8907499);
        double r8907503 = sin(r8907497);
        double r8907504 = 16.0;
        double r8907505 = r8907503 / r8907504;
        double r8907506 = r8907502 - r8907505;
        double r8907507 = r8907502 / r8907504;
        double r8907508 = r8907503 - r8907507;
        double r8907509 = 2.0;
        double r8907510 = sqrt(r8907509);
        double r8907511 = r8907508 * r8907510;
        double r8907512 = r8907506 * r8907511;
        double r8907513 = r8907501 * r8907512;
        double r8907514 = exp(r8907513);
        double r8907515 = log(r8907514);
        double r8907516 = r8907515 + r8907509;
        double r8907517 = 3.0;
        double r8907518 = r8907517 * r8907517;
        double r8907519 = 5.0;
        double r8907520 = r8907518 - r8907519;
        double r8907521 = sqrt(r8907519);
        double r8907522 = r8907517 + r8907521;
        double r8907523 = r8907520 / r8907522;
        double r8907524 = r8907523 / r8907509;
        double r8907525 = r8907500 * r8907524;
        double r8907526 = 1.0;
        double r8907527 = r8907521 - r8907526;
        double r8907528 = r8907527 / r8907509;
        double r8907529 = r8907498 * r8907528;
        double r8907530 = r8907529 + r8907526;
        double r8907531 = r8907525 + r8907530;
        double r8907532 = r8907531 * r8907517;
        double r8907533 = r8907516 / r8907532;
        return r8907533;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 flip--0.5

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

  4. Simplified0.5

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

  6. Applied add-log-exp0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +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))))))