Average Error: 39.8 → 0.7
Time: 13.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r60488 = x;
        double r60489 = eps;
        double r60490 = r60488 + r60489;
        double r60491 = cos(r60490);
        double r60492 = cos(r60488);
        double r60493 = r60491 - r60492;
        return r60493;
}
↓
double f(double x, double eps) {
        double r60494 = eps;
        double r60495 = -7.455928426940857e-05;
        bool r60496 = r60494 <= r60495;
        double r60497 = 9.949689884521833e-06;
        bool r60498 = r60494 <= r60497;
        double r60499 = !r60498;
        bool r60500 = r60496 || r60499;
        double r60501 = x;
        double r60502 = cos(r60501);
        double r60503 = cos(r60494);
        double r60504 = r60502 * r60503;
        double r60505 = sin(r60501);
        double r60506 = sin(r60494);
        double r60507 = r60505 * r60506;
        double r60508 = r60504 - r60507;
        double r60509 = r60508 - r60502;
        double r60510 = -2.0;
        double r60511 = 2.0;
        double r60512 = r60494 / r60511;
        double r60513 = sin(r60512);
        double r60514 = r60501 + r60494;
        double r60515 = r60514 + r60501;
        double r60516 = r60515 / r60511;
        double r60517 = sin(r60516);
        double r60518 = r60513 * r60517;
        double r60519 = r60510 * r60518;
        double r60520 = r60500 ? r60509 : r60519;
        return r60520;
}

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 < -7.455928426940857e-05 or 9.949689884521833e-06 < eps

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -7.455928426940857e-05 < eps < 9.949689884521833e-06

    1. Initial program 49.6

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

    3. Applied diff-cos38.0

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

    4. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))