Average Error: 0.5 → 0.5
Time: 17.7s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{1}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y}{\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3}}}\]
\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{1}{\frac{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y}{\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}}}
double f(double x, double y) {
        double r212361 = 2.0;
        double r212362 = sqrt(r212361);
        double r212363 = x;
        double r212364 = sin(r212363);
        double r212365 = y;
        double r212366 = sin(r212365);
        double r212367 = 16.0;
        double r212368 = r212366 / r212367;
        double r212369 = r212364 - r212368;
        double r212370 = r212362 * r212369;
        double r212371 = r212364 / r212367;
        double r212372 = r212366 - r212371;
        double r212373 = r212370 * r212372;
        double r212374 = cos(r212363);
        double r212375 = cos(r212365);
        double r212376 = r212374 - r212375;
        double r212377 = r212373 * r212376;
        double r212378 = r212361 + r212377;
        double r212379 = 3.0;
        double r212380 = 1.0;
        double r212381 = 5.0;
        double r212382 = sqrt(r212381);
        double r212383 = r212382 - r212380;
        double r212384 = r212383 / r212361;
        double r212385 = r212384 * r212374;
        double r212386 = r212380 + r212385;
        double r212387 = r212379 - r212382;
        double r212388 = r212387 / r212361;
        double r212389 = r212388 * r212375;
        double r212390 = r212386 + r212389;
        double r212391 = r212379 * r212390;
        double r212392 = r212378 / r212391;
        return r212392;
}
↓
double f(double x, double y) {
        double r212393 = 1.0;
        double r212394 = 1.0;
        double r212395 = 5.0;
        double r212396 = sqrt(r212395);
        double r212397 = r212396 - r212394;
        double r212398 = 2.0;
        double r212399 = r212397 / r212398;
        double r212400 = x;
        double r212401 = cos(r212400);
        double r212402 = r212399 * r212401;
        double r212403 = r212394 + r212402;
        double r212404 = 3.0;
        double r212405 = r212404 * r212404;
        double r212406 = -r212395;
        double r212407 = r212405 + r212406;
        double r212408 = r212404 + r212396;
        double r212409 = r212407 / r212408;
        double r212410 = r212409 / r212398;
        double r212411 = y;
        double r212412 = cos(r212411);
        double r212413 = r212410 * r212412;
        double r212414 = r212403 + r212413;
        double r212415 = sqrt(r212398);
        double r212416 = sin(r212400);
        double r212417 = sin(r212411);
        double r212418 = 16.0;
        double r212419 = r212417 / r212418;
        double r212420 = r212416 - r212419;
        double r212421 = r212415 * r212420;
        double r212422 = r212416 / r212418;
        double r212423 = r212417 - r212422;
        double r212424 = r212421 * r212423;
        double r212425 = r212401 - r212412;
        double r212426 = r212424 * r212425;
        double r212427 = r212398 + r212426;
        double r212428 = r212427 / r212404;
        double r212429 = r212414 / r212428;
        double r212430 = r212393 / r212429;
        return r212430;
}

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

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

  6. Applied associate-/r*0.4

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

  8. Applied clear-num0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020083 
(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))))))