Average Error: 39.7 → 0.8
Time: 36.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5070320787486625 \cdot 10^{-08}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 6.268153124227172 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.5070320787486625 \cdot 10^{-08}:\\
\;\;\;\;\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 6.268153124227172 \cdot 10^{-08}:\\
\;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r6542044 = x;
        double r6542045 = eps;
        double r6542046 = r6542044 + r6542045;
        double r6542047 = cos(r6542046);
        double r6542048 = cos(r6542044);
        double r6542049 = r6542047 - r6542048;
        return r6542049;
}
↓
double f(double x, double eps) {
        double r6542050 = eps;
        double r6542051 = -1.5070320787486625e-08;
        bool r6542052 = r6542050 <= r6542051;
        double r6542053 = x;
        double r6542054 = cos(r6542053);
        double r6542055 = cos(r6542050);
        double r6542056 = r6542054 * r6542055;
        double r6542057 = sin(r6542050);
        double r6542058 = sin(r6542053);
        double r6542059 = fma(r6542057, r6542058, r6542054);
        double r6542060 = r6542056 - r6542059;
        double r6542061 = 6.268153124227172e-08;
        bool r6542062 = r6542050 <= r6542061;
        double r6542063 = 2.0;
        double r6542064 = r6542050 / r6542063;
        double r6542065 = sin(r6542064);
        double r6542066 = -2.0;
        double r6542067 = r6542053 + r6542050;
        double r6542068 = r6542067 + r6542053;
        double r6542069 = r6542068 / r6542063;
        double r6542070 = sin(r6542069);
        double r6542071 = log1p(r6542070);
        double r6542072 = expm1(r6542071);
        double r6542073 = r6542066 * r6542072;
        double r6542074 = r6542065 * r6542073;
        double r6542075 = r6542062 ? r6542074 : r6542060;
        double r6542076 = r6542052 ? r6542060 : r6542075;
        return r6542076;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.5070320787486625e-08 or 6.268153124227172e-08 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum1.1

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

    4. Applied associate--l-1.1

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

    5. Simplified1.1

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

    if -1.5070320787486625e-08 < eps < 6.268153124227172e-08

    1. Initial program 48.9

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

    3. Applied diff-cos37.8

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

    4. Simplified0.3

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

    6. Applied associate-*r*0.3

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

    8. Applied expm1-log1p-u0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5070320787486625 \cdot 10^{-08}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 6.268153124227172 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))