Average Error: 39.5 → 0.8
Time: 26.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.9016104204497692 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.05585815658226044:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*))} - 1)^* \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.9016104204497692 \cdot 10^{-06}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.05585815658226044:\\
\;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*))} - 1)^* \cdot -2\\

\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 r2006341 = x;
        double r2006342 = eps;
        double r2006343 = r2006341 + r2006342;
        double r2006344 = cos(r2006343);
        double r2006345 = cos(r2006341);
        double r2006346 = r2006344 - r2006345;
        return r2006346;
}
↓
double f(double x, double eps) {
        double r2006347 = eps;
        double r2006348 = -1.9016104204497692e-06;
        bool r2006349 = r2006347 <= r2006348;
        double r2006350 = x;
        double r2006351 = cos(r2006350);
        double r2006352 = cos(r2006347);
        double r2006353 = r2006351 * r2006352;
        double r2006354 = sin(r2006350);
        double r2006355 = sin(r2006347);
        double r2006356 = r2006354 * r2006355;
        double r2006357 = r2006353 - r2006356;
        double r2006358 = r2006357 - r2006351;
        double r2006359 = 0.05585815658226044;
        bool r2006360 = r2006347 <= r2006359;
        double r2006361 = 2.0;
        double r2006362 = r2006347 / r2006361;
        double r2006363 = sin(r2006362);
        double r2006364 = r2006350 + r2006347;
        double r2006365 = r2006364 + r2006350;
        double r2006366 = r2006365 / r2006361;
        double r2006367 = sin(r2006366);
        double r2006368 = expm1(r2006367);
        double r2006369 = log1p(r2006368);
        double r2006370 = r2006363 * r2006369;
        double r2006371 = log1p(r2006370);
        double r2006372 = expm1(r2006371);
        double r2006373 = -2.0;
        double r2006374 = r2006372 * r2006373;
        double r2006375 = r2006360 ? r2006374 : r2006358;
        double r2006376 = r2006349 ? r2006358 : r2006375;
        return r2006376;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.9016104204497692e-06 or 0.05585815658226044 < eps

    1. Initial program 29.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -1.9016104204497692e-06 < eps < 0.05585815658226044

    1. Initial program 49.6

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

    3. Applied diff-cos38.2

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

    4. Simplified0.6

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

    6. Applied log1p-expm1-u0.7

      \[\leadsto -2 \cdot \left(\color{blue}{\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
    7. Using strategy rm

    8. Applied expm1-log1p-u0.7

      \[\leadsto -2 \cdot \color{blue}{(e^{\log_* (1 + \log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right))} - 1)^*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.9016104204497692 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.05585815658226044:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*))} - 1)^* \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))