Average Error: 39.6 → 0.7
Time: 14.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 1.2384991687240798 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 1.2384991687240798 \cdot 10^{-5}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\

\end{array}
double f(double x, double eps) {
        double r49385 = x;
        double r49386 = eps;
        double r49387 = r49385 + r49386;
        double r49388 = cos(r49387);
        double r49389 = cos(r49385);
        double r49390 = r49388 - r49389;
        return r49390;
}
↓
double f(double x, double eps) {
        double r49391 = eps;
        double r49392 = -0.0002414674700011765;
        bool r49393 = r49391 <= r49392;
        double r49394 = 1.2384991687240798e-05;
        bool r49395 = r49391 <= r49394;
        double r49396 = !r49395;
        bool r49397 = r49393 || r49396;
        double r49398 = x;
        double r49399 = cos(r49398);
        double r49400 = cos(r49391);
        double r49401 = r49399 * r49400;
        double r49402 = sin(r49398);
        double r49403 = sin(r49391);
        double r49404 = r49402 * r49403;
        double r49405 = r49401 - r49404;
        double r49406 = r49405 - r49399;
        double r49407 = -2.0;
        double r49408 = 2.0;
        double r49409 = r49391 / r49408;
        double r49410 = sin(r49409);
        double r49411 = r49407 * r49410;
        double r49412 = r49398 + r49391;
        double r49413 = r49412 + r49398;
        double r49414 = r49413 / r49408;
        double r49415 = sin(r49414);
        double r49416 = r49411 * r49415;
        double r49417 = r49397 ? r49406 : r49416;
        return r49417;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.0002414674700011765 or 1.2384991687240798e-05 < eps

    1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]

    if -0.0002414674700011765 < eps < 1.2384991687240798e-05

    1. Initial program 48.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied diff-cos37.9

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

    4. Simplified0.5

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied associate-*r*0.5

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 1.2384991687240798 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))