Average Error: 39.8 → 0.9
Time: 23.7s
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 r1932583 = x;
        double r1932584 = eps;
        double r1932585 = r1932583 + r1932584;
        double r1932586 = cos(r1932585);
        double r1932587 = cos(r1932583);
        double r1932588 = r1932586 - r1932587;
        return r1932588;
}
↓
double f(double x, double eps) {
        double r1932589 = eps;
        double r1932590 = -65.72641980047264;
        bool r1932591 = r1932589 <= r1932590;
        double r1932592 = x;
        double r1932593 = cos(r1932592);
        double r1932594 = cos(r1932589);
        double r1932595 = r1932593 * r1932594;
        double r1932596 = sin(r1932592);
        double r1932597 = sin(r1932589);
        double r1932598 = r1932596 * r1932597;
        double r1932599 = r1932593 + r1932598;
        double r1932600 = r1932595 - r1932599;
        double r1932601 = 0.010182833365905121;
        bool r1932602 = r1932589 <= r1932601;
        double r1932603 = -2.0;
        double r1932604 = 2.0;
        double r1932605 = r1932589 / r1932604;
        double r1932606 = sin(r1932605);
        double r1932607 = r1932592 + r1932589;
        double r1932608 = r1932607 + r1932592;
        double r1932609 = r1932608 / r1932604;
        double r1932610 = sin(r1932609);
        double r1932611 = r1932606 * r1932610;
        double r1932612 = r1932603 * r1932611;
        double r1932613 = r1932595 - r1932598;
        double r1932614 = r1932613 - r1932593;
        double r1932615 = r1932602 ? r1932612 : r1932614;
        double r1932616 = r1932591 ? r1932600 : r1932615;
        return r1932616;
}

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 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))