Average Error: 39.3 → 0.7
Time: 20.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.8863753739128075 \cdot 10^{-06}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.8863753739128075 \cdot 10^{-06}:\\
\;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r1561612 = x;
        double r1561613 = eps;
        double r1561614 = r1561612 + r1561613;
        double r1561615 = cos(r1561614);
        double r1561616 = cos(r1561612);
        double r1561617 = r1561615 - r1561616;
        return r1561617;
}
↓
double f(double x, double eps) {
        double r1561618 = eps;
        double r1561619 = -0.0006966600469417059;
        bool r1561620 = r1561618 <= r1561619;
        double r1561621 = x;
        double r1561622 = cos(r1561621);
        double r1561623 = cos(r1561618);
        double r1561624 = r1561622 * r1561623;
        double r1561625 = sin(r1561621);
        double r1561626 = sin(r1561618);
        double r1561627 = r1561625 * r1561626;
        double r1561628 = r1561624 - r1561627;
        double r1561629 = r1561628 - r1561622;
        double r1561630 = 2.8863753739128075e-06;
        bool r1561631 = r1561618 <= r1561630;
        double r1561632 = r1561621 + r1561618;
        double r1561633 = r1561621 + r1561632;
        double r1561634 = 2.0;
        double r1561635 = r1561633 / r1561634;
        double r1561636 = sin(r1561635);
        double r1561637 = -2.0;
        double r1561638 = r1561618 / r1561634;
        double r1561639 = sin(r1561638);
        double r1561640 = r1561637 * r1561639;
        double r1561641 = r1561636 * r1561640;
        double r1561642 = r1561622 + r1561627;
        double r1561643 = r1561624 - r1561642;
        double r1561644 = r1561631 ? r1561641 : r1561643;
        double r1561645 = r1561620 ? r1561629 : r1561644;
        return r1561645;
}

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 < -0.0006966600469417059

    1. Initial program 29.1

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

    3. Applied cos-sum0.8

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

    if -0.0006966600469417059 < eps < 2.8863753739128075e-06

    1. Initial program 49.2

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

    3. Applied diff-cos37.3

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

    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 2.8863753739128075e-06 < eps

    1. Initial program 30.2

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.8863753739128075 \cdot 10^{-06}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))