Average Error: 39.8 → 0.7
Time: 26.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.870955514870532 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.0002018639648153477:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.870955514870532 \cdot 10^{-06}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.0002018639648153477:\\
\;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot -2\\

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

\end{array}
double f(double x, double eps) {
        double r2413520 = x;
        double r2413521 = eps;
        double r2413522 = r2413520 + r2413521;
        double r2413523 = cos(r2413522);
        double r2413524 = cos(r2413520);
        double r2413525 = r2413523 - r2413524;
        return r2413525;
}
↓
double f(double x, double eps) {
        double r2413526 = eps;
        double r2413527 = -9.870955514870532e-06;
        bool r2413528 = r2413526 <= r2413527;
        double r2413529 = x;
        double r2413530 = cos(r2413529);
        double r2413531 = cos(r2413526);
        double r2413532 = r2413530 * r2413531;
        double r2413533 = sin(r2413529);
        double r2413534 = sin(r2413526);
        double r2413535 = r2413533 * r2413534;
        double r2413536 = r2413532 - r2413535;
        double r2413537 = r2413536 - r2413530;
        double r2413538 = 0.0002018639648153477;
        bool r2413539 = r2413526 <= r2413538;
        double r2413540 = 2.0;
        double r2413541 = r2413526 / r2413540;
        double r2413542 = sin(r2413541);
        double r2413543 = r2413529 + r2413526;
        double r2413544 = r2413543 + r2413529;
        double r2413545 = r2413544 / r2413540;
        double r2413546 = sin(r2413545);
        double r2413547 = log1p(r2413546);
        double r2413548 = expm1(r2413547);
        double r2413549 = r2413542 * r2413548;
        double r2413550 = log1p(r2413549);
        double r2413551 = expm1(r2413550);
        double r2413552 = -2.0;
        double r2413553 = r2413551 * r2413552;
        double r2413554 = r2413539 ? r2413553 : r2413537;
        double r2413555 = r2413528 ? r2413537 : r2413554;
        return r2413555;
}

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 < -9.870955514870532e-06 or 0.0002018639648153477 < eps

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -9.870955514870532e-06 < eps < 0.0002018639648153477

    1. Initial program 49.5

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

    3. Applied diff-cos38.1

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

    4. Simplified0.5

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

    6. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\right)\right)}\]
    7. Using strategy rm

    8. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.870955514870532 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.0002018639648153477:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))