Average Error: 39.4 → 0.7
Time: 18.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.347365952010613 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.2247927857601395 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.347365952010613 \cdot 10^{-07}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.2247927857601395 \cdot 10^{-06}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{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 r408365 = x;
        double r408366 = eps;
        double r408367 = r408365 + r408366;
        double r408368 = cos(r408367);
        double r408369 = cos(r408365);
        double r408370 = r408368 - r408369;
        return r408370;
}
↓
double f(double x, double eps) {
        double r408371 = eps;
        double r408372 = -1.347365952010613e-07;
        bool r408373 = r408371 <= r408372;
        double r408374 = x;
        double r408375 = cos(r408374);
        double r408376 = cos(r408371);
        double r408377 = r408375 * r408376;
        double r408378 = sin(r408374);
        double r408379 = sin(r408371);
        double r408380 = r408378 * r408379;
        double r408381 = r408377 - r408380;
        double r408382 = r408381 - r408375;
        double r408383 = 2.2247927857601395e-06;
        bool r408384 = r408371 <= r408383;
        double r408385 = -2.0;
        double r408386 = 2.0;
        double r408387 = r408371 / r408386;
        double r408388 = sin(r408387);
        double r408389 = r408374 + r408371;
        double r408390 = r408389 + r408374;
        double r408391 = r408390 / r408386;
        double r408392 = sin(r408391);
        double r408393 = r408388 * r408392;
        double r408394 = r408385 * r408393;
        double r408395 = r408384 ? r408394 : r408382;
        double r408396 = r408373 ? r408382 : r408395;
        return r408396;
}

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 < -1.347365952010613e-07 or 2.2247927857601395e-06 < eps

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -1.347365952010613e-07 < eps < 2.2247927857601395e-06

    1. Initial program 48.9

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

    3. Applied diff-cos38.1

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

    4. Simplified0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.347365952010613 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.2247927857601395 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))