Average Error: 39.8 → 0.8
Time: 25.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.59746672523127812 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 2.86080349302899809 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.59746672523127812 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 2.86080349302899809 \cdot 10^{-8}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r22578 = x;
        double r22579 = eps;
        double r22580 = r22578 + r22579;
        double r22581 = cos(r22580);
        double r22582 = cos(r22578);
        double r22583 = r22581 - r22582;
        return r22583;
}
↓
double f(double x, double eps) {
        double r22584 = eps;
        double r22585 = -3.597466725231278e-05;
        bool r22586 = r22584 <= r22585;
        double r22587 = 2.860803493028998e-08;
        bool r22588 = r22584 <= r22587;
        double r22589 = !r22588;
        bool r22590 = r22586 || r22589;
        double r22591 = x;
        double r22592 = cos(r22591);
        double r22593 = cos(r22584);
        double r22594 = r22592 * r22593;
        double r22595 = sin(r22591);
        double r22596 = sin(r22584);
        double r22597 = r22595 * r22596;
        double r22598 = r22597 + r22592;
        double r22599 = r22594 - r22598;
        double r22600 = -2.0;
        double r22601 = 2.0;
        double r22602 = r22584 / r22601;
        double r22603 = sin(r22602);
        double r22604 = r22591 + r22584;
        double r22605 = r22604 + r22591;
        double r22606 = r22605 / r22601;
        double r22607 = sin(r22606);
        double r22608 = r22603 * r22607;
        double r22609 = r22600 * r22608;
        double r22610 = r22590 ? r22599 : r22609;
        return r22610;
}

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 < -3.597466725231278e-05 or 2.860803493028998e-08 < eps

    1. Initial program 30.4

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.0

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

    if -3.597466725231278e-05 < eps < 2.860803493028998e-08

    1. Initial program 49.5

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

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.59746672523127812 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 2.86080349302899809 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))