Average Error: 39.8 → 0.9
Time: 20.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -65.72641980047264:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.010182833365905121:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -65.72641980047264:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \le 0.010182833365905121:\\
\;\;\;\;-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 r4126414 = x;
        double r4126415 = eps;
        double r4126416 = r4126414 + r4126415;
        double r4126417 = cos(r4126416);
        double r4126418 = cos(r4126414);
        double r4126419 = r4126417 - r4126418;
        return r4126419;
}
↓
double f(double x, double eps) {
        double r4126420 = eps;
        double r4126421 = -65.72641980047264;
        bool r4126422 = r4126420 <= r4126421;
        double r4126423 = x;
        double r4126424 = cos(r4126423);
        double r4126425 = cos(r4126420);
        double r4126426 = r4126424 * r4126425;
        double r4126427 = sin(r4126423);
        double r4126428 = sin(r4126420);
        double r4126429 = r4126427 * r4126428;
        double r4126430 = r4126424 + r4126429;
        double r4126431 = r4126426 - r4126430;
        double r4126432 = 0.010182833365905121;
        bool r4126433 = r4126420 <= r4126432;
        double r4126434 = -2.0;
        double r4126435 = 2.0;
        double r4126436 = r4126420 / r4126435;
        double r4126437 = sin(r4126436);
        double r4126438 = r4126423 + r4126420;
        double r4126439 = r4126438 + r4126423;
        double r4126440 = r4126439 / r4126435;
        double r4126441 = sin(r4126440);
        double r4126442 = r4126437 * r4126441;
        double r4126443 = r4126434 * r4126442;
        double r4126444 = r4126426 - r4126429;
        double r4126445 = r4126444 - r4126424;
        double r4126446 = r4126433 ? r4126443 : r4126445;
        double r4126447 = r4126422 ? r4126431 : r4126446;
        return r4126447;
}

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 < -65.72641980047264

    1. Initial program 29.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-0.9

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

    if -65.72641980047264 < eps < 0.010182833365905121

    1. Initial program 49.1

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

    3. Applied diff-cos37.8

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

    4. Simplified0.9

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

    if 0.010182833365905121 < eps

    1. Initial program 31.0

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -65.72641980047264:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.010182833365905121:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))