Average Error: 39.1 → 0.8
Time: 30.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\right)\\

\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 r19709 = x;
        double r19710 = eps;
        double r19711 = r19709 + r19710;
        double r19712 = cos(r19711);
        double r19713 = cos(r19709);
        double r19714 = r19712 - r19713;
        return r19714;
}
↓
double f(double x, double eps) {
        double r19715 = eps;
        double r19716 = -0.00026844987953330624;
        bool r19717 = r19715 <= r19716;
        double r19718 = x;
        double r19719 = cos(r19718);
        double r19720 = cos(r19715);
        double r19721 = r19719 * r19720;
        double r19722 = sin(r19718);
        double r19723 = sin(r19715);
        double r19724 = fma(r19722, r19723, r19719);
        double r19725 = r19721 - r19724;
        double r19726 = 3.028675491606891e-06;
        bool r19727 = r19715 <= r19726;
        double r19728 = -2.0;
        double r19729 = 2.0;
        double r19730 = r19715 / r19729;
        double r19731 = sin(r19730);
        double r19732 = fma(r19729, r19718, r19715);
        double r19733 = r19732 / r19729;
        double r19734 = sin(r19733);
        double r19735 = expm1(r19734);
        double r19736 = log1p(r19735);
        double r19737 = r19731 * r19736;
        double r19738 = r19728 * r19737;
        double r19739 = r19722 * r19723;
        double r19740 = r19721 - r19739;
        double r19741 = r19740 - r19719;
        double r19742 = r19727 ? r19738 : r19741;
        double r19743 = r19717 ? r19725 : r19742;
        return r19743;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00026844987953330624

    1. Initial program 29.2

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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

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

    if -0.00026844987953330624 < eps < 3.028675491606891e-06

    1. Initial program 49.2

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

    3. Applied diff-cos38.3

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

    4. Simplified0.5

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

    6. Applied log1p-expm1-u0.6

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

    if 3.028675491606891e-06 < eps

    1. Initial program 29.6

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

    3. Applied cos-sum1.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))