Average Error: 39.6 → 0.7
Time: 17.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r53575 = x;
        double r53576 = eps;
        double r53577 = r53575 + r53576;
        double r53578 = cos(r53577);
        double r53579 = cos(r53575);
        double r53580 = r53578 - r53579;
        return r53580;
}
↓
double f(double x, double eps) {
        double r53581 = eps;
        double r53582 = -0.0002414674700011765;
        bool r53583 = r53581 <= r53582;
        double r53584 = x;
        double r53585 = cos(r53584);
        double r53586 = cos(r53581);
        double r53587 = r53585 * r53586;
        double r53588 = sin(r53584);
        double r53589 = sin(r53581);
        double r53590 = r53588 * r53589;
        double r53591 = r53587 - r53590;
        double r53592 = r53591 - r53585;
        double r53593 = 6.808328837524803e-05;
        bool r53594 = r53581 <= r53593;
        double r53595 = -2.0;
        double r53596 = 2.0;
        double r53597 = r53581 / r53596;
        double r53598 = sin(r53597);
        double r53599 = r53595 * r53598;
        double r53600 = fma(r53596, r53584, r53581);
        double r53601 = r53600 / r53596;
        double r53602 = sin(r53601);
        double r53603 = r53599 * r53602;
        double r53604 = expm1(r53603);
        double r53605 = log1p(r53604);
        double r53606 = fma(r53588, r53589, r53585);
        double r53607 = r53587 - r53606;
        double r53608 = r53594 ? r53605 : r53607;
        double r53609 = r53583 ? r53592 : r53608;
        return r53609;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.0002414674700011765

    1. Initial program 31.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.0002414674700011765 < eps < 6.808328837524803e-05

    1. Initial program 48.9

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

    3. Applied diff-cos37.9

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

    4. Simplified0.5

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

    6. Applied *-un-lft-identity0.5

      \[\leadsto \color{blue}{\left(1 \cdot -2\right)} \cdot \left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\]

    7. Applied associate-*l*0.5

      \[\leadsto \color{blue}{1 \cdot \left(-2 \cdot \left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\]

    8. Simplified0.5

      \[\leadsto 1 \cdot \color{blue}{\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
    9. Using strategy rm

    10. Applied log1p-expm1-u0.5

      \[\leadsto 1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\]

    if 6.808328837524803e-05 < eps

    1. Initial program 29.6

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

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))