Average Error: 40.1 → 0.7
Time: 19.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\
\;\;\;\;\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 r1839481 = x;
        double r1839482 = eps;
        double r1839483 = r1839481 + r1839482;
        double r1839484 = cos(r1839483);
        double r1839485 = cos(r1839481);
        double r1839486 = r1839484 - r1839485;
        return r1839486;
}
↓
double f(double x, double eps) {
        double r1839487 = eps;
        double r1839488 = -1.4689161588736088e-05;
        bool r1839489 = r1839487 <= r1839488;
        double r1839490 = x;
        double r1839491 = cos(r1839490);
        double r1839492 = cos(r1839487);
        double r1839493 = r1839491 * r1839492;
        double r1839494 = sin(r1839490);
        double r1839495 = sin(r1839487);
        double r1839496 = r1839494 * r1839495;
        double r1839497 = r1839491 + r1839496;
        double r1839498 = r1839493 - r1839497;
        double r1839499 = 0.0004985547184553873;
        bool r1839500 = r1839487 <= r1839499;
        double r1839501 = r1839490 + r1839487;
        double r1839502 = r1839490 + r1839501;
        double r1839503 = 2.0;
        double r1839504 = r1839502 / r1839503;
        double r1839505 = sin(r1839504);
        double r1839506 = -2.0;
        double r1839507 = r1839487 / r1839503;
        double r1839508 = sin(r1839507);
        double r1839509 = r1839506 * r1839508;
        double r1839510 = r1839505 * r1839509;
        double r1839511 = r1839493 - r1839496;
        double r1839512 = r1839511 - r1839491;
        double r1839513 = r1839500 ? r1839510 : r1839512;
        double r1839514 = r1839489 ? r1839498 : r1839513;
        return r1839514;
}

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 < -1.4689161588736088e-05

    1. Initial program 29.8

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

    if -1.4689161588736088e-05 < eps < 0.0004985547184553873

    1. Initial program 49.3

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

    3. Applied diff-cos37.6

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

    4. Simplified0.5

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

    6. Applied associate-*r*0.5

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

    if 0.0004985547184553873 < eps

    1. Initial program 32.1

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

    3. Applied cos-sum0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4689161588736088 \cdot 10^{-05}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.0004985547184553873:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))