Average Error: 40.0 → 0.7
Time: 19.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\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 r335599 = x;
        double r335600 = eps;
        double r335601 = r335599 + r335600;
        double r335602 = cos(r335601);
        double r335603 = cos(r335599);
        double r335604 = r335602 - r335603;
        return r335604;
}
↓
double f(double x, double eps) {
        double r335605 = eps;
        double r335606 = -9.798283548084824e-06;
        bool r335607 = r335605 <= r335606;
        double r335608 = x;
        double r335609 = cos(r335608);
        double r335610 = cos(r335605);
        double r335611 = r335609 * r335610;
        double r335612 = sin(r335608);
        double r335613 = sin(r335605);
        double r335614 = r335612 * r335613;
        double r335615 = r335611 - r335614;
        double r335616 = r335615 - r335609;
        double r335617 = 0.00011275979133792468;
        bool r335618 = r335605 <= r335617;
        double r335619 = 2.0;
        double r335620 = fma(r335619, r335608, r335605);
        double r335621 = r335620 / r335619;
        double r335622 = sin(r335621);
        double r335623 = log1p(r335622);
        double r335624 = expm1(r335623);
        double r335625 = -2.0;
        double r335626 = r335605 / r335619;
        double r335627 = sin(r335626);
        double r335628 = r335625 * r335627;
        double r335629 = r335624 * r335628;
        double r335630 = r335618 ? r335629 : r335616;
        double r335631 = r335607 ? r335616 : r335630;
        return r335631;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.798283548084824e-06 or 0.00011275979133792468 < eps

    1. Initial program 30.7

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

    3. Applied cos-sum1.0

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

    if -9.798283548084824e-06 < eps < 0.00011275979133792468

    1. Initial program 49.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.4

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

    6. Applied associate-*r*0.4

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

    8. Applied expm1-log1p-u0.5

      \[\leadsto \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))