Average Error: 39.1 → 0.8
Time: 15.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.03196418493759398127274096168548567220569:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 9.763543264731029431170022947483744246711 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.03196418493759398127274096168548567220569:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 9.763543264731029431170022947483744246711 \cdot 10^{-5}:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r20431 = x;
        double r20432 = eps;
        double r20433 = r20431 + r20432;
        double r20434 = cos(r20433);
        double r20435 = cos(r20431);
        double r20436 = r20434 - r20435;
        return r20436;
}
↓
double f(double x, double eps) {
        double r20437 = eps;
        double r20438 = -0.03196418493759398;
        bool r20439 = r20437 <= r20438;
        double r20440 = x;
        double r20441 = cos(r20440);
        double r20442 = cos(r20437);
        double r20443 = r20441 * r20442;
        double r20444 = sin(r20440);
        double r20445 = sin(r20437);
        double r20446 = fma(r20444, r20445, r20441);
        double r20447 = r20443 - r20446;
        double r20448 = 9.76354326473103e-05;
        bool r20449 = r20437 <= r20448;
        double r20450 = -2.0;
        double r20451 = 2.0;
        double r20452 = r20437 / r20451;
        double r20453 = sin(r20452);
        double r20454 = r20450 * r20453;
        double r20455 = r20440 + r20437;
        double r20456 = r20455 + r20440;
        double r20457 = r20456 / r20451;
        double r20458 = sin(r20457);
        double r20459 = expm1(r20458);
        double r20460 = log1p(r20459);
        double r20461 = r20454 * r20460;
        double r20462 = r20442 * r20441;
        double r20463 = r20444 * r20445;
        double r20464 = r20462 - r20463;
        double r20465 = r20464 - r20441;
        double r20466 = r20449 ? r20461 : r20465;
        double r20467 = r20439 ? r20447 : r20466;
        return r20467;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.03196418493759398

    1. Initial program 29.8

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

    5. Simplified0.8

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

    if -0.03196418493759398 < eps < 9.76354326473103e-05

    1. Initial program 49.1

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

    3. Applied diff-cos38.2

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

    4. Simplified0.6

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied associate-*r*0.6

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon + 0}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)}\]

    7. Simplified0.6

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

    9. Applied log1p-expm1-u0.7

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

    if 9.76354326473103e-05 < eps

    1. Initial program 29.1

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

    3. Applied cos-sum0.9

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

    4. Simplified0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.03196418493759398127274096168548567220569:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 9.763543264731029431170022947483744246711 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))