Average Error: 39.8 → 0.7
Time: 21.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\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 r21376 = x;
        double r21377 = eps;
        double r21378 = r21376 + r21377;
        double r21379 = cos(r21378);
        double r21380 = cos(r21376);
        double r21381 = r21379 - r21380;
        return r21381;
}
↓
double f(double x, double eps) {
        double r21382 = eps;
        double r21383 = -0.00815188689103541;
        bool r21384 = r21382 <= r21383;
        double r21385 = x;
        double r21386 = cos(r21385);
        double r21387 = cos(r21382);
        double r21388 = r21386 * r21387;
        double r21389 = sin(r21385);
        double r21390 = sin(r21382);
        double r21391 = r21389 * r21390;
        double r21392 = r21391 + r21386;
        double r21393 = r21388 - r21392;
        double r21394 = 0.04140103919524168;
        bool r21395 = r21382 <= r21394;
        double r21396 = -2.0;
        double r21397 = 2.0;
        double r21398 = r21382 / r21397;
        double r21399 = sin(r21398);
        double r21400 = r21396 * r21399;
        double r21401 = r21385 + r21382;
        double r21402 = r21401 + r21385;
        double r21403 = r21402 / r21397;
        double r21404 = sin(r21403);
        double r21405 = r21400 * r21404;
        double r21406 = r21388 - r21391;
        double r21407 = r21406 - r21386;
        double r21408 = r21395 ? r21405 : r21407;
        double r21409 = r21384 ? r21393 : r21408;
        return r21409;
}

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 3 regimes

  2. if eps < -0.00815188689103541

    1. Initial program 30.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Applied associate--l-0.8

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

    if -0.00815188689103541 < eps < 0.04140103919524168

    1. Initial program 48.8

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

    3. Applied diff-cos37.0

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

    4. Simplified0.6

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

    6. Applied associate-*r*0.6

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

    if 0.04140103919524168 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.04140103919524167758181576459719508420676:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))