Average Error: 39.8 → 0.7
Time: 37.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\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 r29313 = x;
        double r29314 = eps;
        double r29315 = r29313 + r29314;
        double r29316 = cos(r29315);
        double r29317 = cos(r29313);
        double r29318 = r29316 - r29317;
        return r29318;
}
↓
double f(double x, double eps) {
        double r29319 = eps;
        double r29320 = -0.00815188689103541;
        bool r29321 = r29319 <= r29320;
        double r29322 = x;
        double r29323 = cos(r29322);
        double r29324 = cos(r29319);
        double r29325 = r29323 * r29324;
        double r29326 = sin(r29322);
        double r29327 = sin(r29319);
        double r29328 = fma(r29326, r29327, r29323);
        double r29329 = r29325 - r29328;
        double r29330 = 0.04140103919524168;
        bool r29331 = r29319 <= r29330;
        double r29332 = -2.0;
        double r29333 = 2.0;
        double r29334 = r29319 / r29333;
        double r29335 = sin(r29334);
        double r29336 = r29332 * r29335;
        double r29337 = fma(r29333, r29322, r29319);
        double r29338 = r29337 / r29333;
        double r29339 = sin(r29338);
        double r29340 = r29336 * r29339;
        double r29341 = r29326 * r29327;
        double r29342 = r29325 - r29341;
        double r29343 = r29342 - r29323;
        double r29344 = r29331 ? r29340 : r29343;
        double r29345 = r29321 ? r29329 : r29344;
        return r29345;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00815188689103541

    1. Initial program 30.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.00815188689103541 < eps < 0.04140103919524168

    1. Initial program 48.8

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

    3. Applied diff-cos37.0

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

    4. Simplified0.6

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

    6. Applied associate-*r*0.6

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

    if 0.04140103919524168 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))