Average Error: 40.1 → 0.7
Time: 22.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\

\mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\
\;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r1073687 = x;
        double r1073688 = eps;
        double r1073689 = r1073687 + r1073688;
        double r1073690 = cos(r1073689);
        double r1073691 = cos(r1073687);
        double r1073692 = r1073690 - r1073691;
        return r1073692;
}
↓
double f(double x, double eps) {
        double r1073693 = eps;
        double r1073694 = -1.4689161588736088e-05;
        bool r1073695 = r1073693 <= r1073694;
        double r1073696 = x;
        double r1073697 = cos(r1073696);
        double r1073698 = cos(r1073693);
        double r1073699 = r1073697 * r1073698;
        double r1073700 = sin(r1073693);
        double r1073701 = sin(r1073696);
        double r1073702 = fma(r1073700, r1073701, r1073697);
        double r1073703 = r1073699 - r1073702;
        double r1073704 = 0.0004985547184553873;
        bool r1073705 = r1073693 <= r1073704;
        double r1073706 = -2.0;
        double r1073707 = 2.0;
        double r1073708 = r1073693 / r1073707;
        double r1073709 = sin(r1073708);
        double r1073710 = r1073706 * r1073709;
        double r1073711 = r1073696 + r1073693;
        double r1073712 = r1073711 + r1073696;
        double r1073713 = r1073712 / r1073707;
        double r1073714 = sin(r1073713);
        double r1073715 = r1073710 * r1073714;
        double r1073716 = log1p(r1073715);
        double r1073717 = expm1(r1073716);
        double r1073718 = r1073700 * r1073701;
        double r1073719 = r1073699 - r1073718;
        double r1073720 = r1073719 - r1073697;
        double r1073721 = r1073705 ? r1073717 : r1073720;
        double r1073722 = r1073695 ? r1073703 : r1073721;
        return r1073722;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.4689161588736088e-05

    1. Initial program 29.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Applied associate--l-0.9

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

    5. Simplified0.9

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

    if -1.4689161588736088e-05 < eps < 0.0004985547184553873

    1. Initial program 49.3

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

    3. Applied diff-cos37.6

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

    4. Simplified0.5

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

    6. Applied associate-*r*0.5

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

    8. Applied expm1-log1p-u0.5

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

    if 0.0004985547184553873 < eps

    1. Initial program 32.1

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

    3. Applied cos-sum0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))