Average Error: 40.1 → 0.8
Time: 20.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0987473663124097:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.0143977468052977 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0987473663124097:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.0143977468052977 \cdot 10^{-08}:\\
\;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{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 r1150617 = x;
        double r1150618 = eps;
        double r1150619 = r1150617 + r1150618;
        double r1150620 = cos(r1150619);
        double r1150621 = cos(r1150617);
        double r1150622 = r1150620 - r1150621;
        return r1150622;
}
↓
double f(double x, double eps) {
        double r1150623 = eps;
        double r1150624 = -0.0987473663124097;
        bool r1150625 = r1150623 <= r1150624;
        double r1150626 = x;
        double r1150627 = cos(r1150626);
        double r1150628 = cos(r1150623);
        double r1150629 = r1150627 * r1150628;
        double r1150630 = sin(r1150626);
        double r1150631 = sin(r1150623);
        double r1150632 = r1150630 * r1150631;
        double r1150633 = r1150629 - r1150632;
        double r1150634 = r1150633 - r1150627;
        double r1150635 = 2.0143977468052977e-08;
        bool r1150636 = r1150623 <= r1150635;
        double r1150637 = r1150626 + r1150623;
        double r1150638 = r1150626 + r1150637;
        double r1150639 = 2.0;
        double r1150640 = r1150638 / r1150639;
        double r1150641 = sin(r1150640);
        double r1150642 = -2.0;
        double r1150643 = r1150623 / r1150639;
        double r1150644 = sin(r1150643);
        double r1150645 = r1150642 * r1150644;
        double r1150646 = r1150641 * r1150645;
        double r1150647 = r1150636 ? r1150646 : r1150634;
        double r1150648 = r1150625 ? r1150634 : r1150647;
        return r1150648;
}

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 < -0.0987473663124097 or 2.0143977468052977e-08 < eps

    1. Initial program 30.5

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

    3. Applied cos-sum1.1

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

    if -0.0987473663124097 < eps < 2.0143977468052977e-08

    1. Initial program 49.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.6

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

    6. Applied associate-*r*0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0987473663124097:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.0143977468052977 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))