Average Error: 39.4 → 0.7
Time: 17.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.1114026085301423468765236179933708626777 \lor \neg \left(\varepsilon \le 8.703484289726959091389006661909633066898 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \sin \left(\frac{2 \cdot x + \varepsilon}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.1114026085301423468765236179933708626777 \lor \neg \left(\varepsilon \le 8.703484289726959091389006661909633066898 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\

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

\end{array}
double f(double x, double eps) {
        double r62594 = x;
        double r62595 = eps;
        double r62596 = r62594 + r62595;
        double r62597 = cos(r62596);
        double r62598 = cos(r62594);
        double r62599 = r62597 - r62598;
        return r62599;
}
↓
double f(double x, double eps) {
        double r62600 = eps;
        double r62601 = -0.11140260853014235;
        bool r62602 = r62600 <= r62601;
        double r62603 = 8.703484289726959e-06;
        bool r62604 = r62600 <= r62603;
        double r62605 = !r62604;
        bool r62606 = r62602 || r62605;
        double r62607 = cos(r62600);
        double r62608 = x;
        double r62609 = cos(r62608);
        double r62610 = r62607 * r62609;
        double r62611 = sin(r62608);
        double r62612 = sin(r62600);
        double r62613 = r62611 * r62612;
        double r62614 = r62610 - r62613;
        double r62615 = r62614 - r62609;
        double r62616 = 2.0;
        double r62617 = r62600 / r62616;
        double r62618 = sin(r62617);
        double r62619 = -2.0;
        double r62620 = r62618 * r62619;
        double r62621 = r62616 * r62608;
        double r62622 = r62621 + r62600;
        double r62623 = r62622 / r62616;
        double r62624 = sin(r62623);
        double r62625 = r62620 * r62624;
        double r62626 = r62606 ? r62615 : r62625;
        return r62626;
}

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 < -0.11140260853014235 or 8.703484289726959e-06 < eps

    1. Initial program 30.1

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

    3. Applied cos-sum0.8

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

    4. Simplified0.8

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

    if -0.11140260853014235 < eps < 8.703484289726959e-06

    1. Initial program 48.7

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

    3. Applied diff-cos36.9

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

    4. Simplified0.5

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

    6. Applied *-un-lft-identity0.5

      \[\leadsto \color{blue}{\left(1 \cdot -2\right)} \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

    7. Applied associate-*l*0.5

      \[\leadsto \color{blue}{1 \cdot \left(-2 \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)}\]

    8. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.1114026085301423468765236179933708626777 \lor \neg \left(\varepsilon \le 8.703484289726959091389006661909633066898 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \sin \left(\frac{2 \cdot x + \varepsilon}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))