Average Error: 39.4 → 0.7
Time: 19.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.156576232300930215672520734448980306297 \cdot 10^{-7}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 8.248660376627990132355344732317803391197 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.156576232300930215672520734448980306297 \cdot 10^{-7}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 8.248660376627990132355344732317803391197 \cdot 10^{-5}:\\
\;\;\;\;-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 r4254426 = x;
        double r4254427 = eps;
        double r4254428 = r4254426 + r4254427;
        double r4254429 = cos(r4254428);
        double r4254430 = cos(r4254426);
        double r4254431 = r4254429 - r4254430;
        return r4254431;
}
↓
double f(double x, double eps) {
        double r4254432 = eps;
        double r4254433 = -3.15657623230093e-07;
        bool r4254434 = r4254432 <= r4254433;
        double r4254435 = x;
        double r4254436 = cos(r4254435);
        double r4254437 = cos(r4254432);
        double r4254438 = r4254436 * r4254437;
        double r4254439 = sin(r4254435);
        double r4254440 = sin(r4254432);
        double r4254441 = r4254439 * r4254440;
        double r4254442 = r4254438 - r4254441;
        double r4254443 = r4254442 - r4254436;
        double r4254444 = 8.24866037662799e-05;
        bool r4254445 = r4254432 <= r4254444;
        double r4254446 = -2.0;
        double r4254447 = 2.0;
        double r4254448 = r4254432 / r4254447;
        double r4254449 = sin(r4254448);
        double r4254450 = r4254435 + r4254432;
        double r4254451 = r4254450 + r4254435;
        double r4254452 = r4254451 / r4254447;
        double r4254453 = sin(r4254452);
        double r4254454 = r4254449 * r4254453;
        double r4254455 = r4254446 * r4254454;
        double r4254456 = r4254445 ? r4254455 : r4254443;
        double r4254457 = r4254434 ? r4254443 : r4254456;
        return r4254457;
}

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 < -3.15657623230093e-07 or 8.24866037662799e-05 < eps

    1. Initial program 30.4

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

    3. Applied cos-sum0.9

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

    if -3.15657623230093e-07 < eps < 8.24866037662799e-05

    1. Initial program 48.8

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

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.156576232300930215672520734448980306297 \cdot 10^{-7}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 8.248660376627990132355344732317803391197 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))