Average Error: 38.9 → 0.7
Time: 25.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.8559924841132814 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00686024552453565:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.8559924841132814 \cdot 10^{-05}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.00686024552453565:\\
\;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \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 r2739006 = x;
        double r2739007 = eps;
        double r2739008 = r2739006 + r2739007;
        double r2739009 = cos(r2739008);
        double r2739010 = cos(r2739006);
        double r2739011 = r2739009 - r2739010;
        return r2739011;
}
↓
double f(double x, double eps) {
        double r2739012 = eps;
        double r2739013 = -2.8559924841132814e-05;
        bool r2739014 = r2739012 <= r2739013;
        double r2739015 = x;
        double r2739016 = cos(r2739015);
        double r2739017 = cos(r2739012);
        double r2739018 = r2739016 * r2739017;
        double r2739019 = sin(r2739015);
        double r2739020 = sin(r2739012);
        double r2739021 = r2739019 * r2739020;
        double r2739022 = r2739018 - r2739021;
        double r2739023 = r2739022 - r2739016;
        double r2739024 = 0.00686024552453565;
        bool r2739025 = r2739012 <= r2739024;
        double r2739026 = 2.0;
        double r2739027 = r2739012 / r2739026;
        double r2739028 = sin(r2739027);
        double r2739029 = -2.0;
        double r2739030 = r2739015 + r2739012;
        double r2739031 = r2739030 + r2739015;
        double r2739032 = r2739031 / r2739026;
        double r2739033 = sin(r2739032);
        double r2739034 = r2739029 * r2739033;
        double r2739035 = r2739028 * r2739034;
        double r2739036 = r2739025 ? r2739035 : r2739023;
        double r2739037 = r2739014 ? r2739023 : r2739036;
        return r2739037;
}

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 < -2.8559924841132814e-05 or 0.00686024552453565 < eps

    1. Initial program 29.8

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

    3. Applied cos-sum0.9

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

    if -2.8559924841132814e-05 < eps < 0.00686024552453565

    1. Initial program 48.4

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

    3. Applied diff-cos36.8

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

    4. Simplified0.5

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

    6. Applied associate-*r*0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.8559924841132814 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00686024552453565:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))