Average Error: 39.5 → 0.4
Time: 22.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\]
double f(double x, double eps) {
        double r2990247 = x;
        double r2990248 = eps;
        double r2990249 = r2990247 + r2990248;
        double r2990250 = cos(r2990249);
        double r2990251 = cos(r2990247);
        double r2990252 = r2990250 - r2990251;
        return r2990252;
}
↓
double f(double x, double eps) {
        double r2990253 = x;
        double r2990254 = cos(r2990253);
        double r2990255 = 0.5;
        double r2990256 = eps;
        double r2990257 = r2990255 * r2990256;
        double r2990258 = sin(r2990257);
        double r2990259 = r2990254 * r2990258;
        double r2990260 = sin(r2990253);
        double r2990261 = cos(r2990257);
        double r2990262 = r2990260 * r2990261;
        double r2990263 = r2990259 + r2990262;
        double r2990264 = r2990263 * r2990258;
        double r2990265 = -2.0;
        double r2990266 = r2990264 * r2990265;
        return r2990266;
}


\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.5

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

  3. Applied diff-cos34.1

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

  4. Simplified15.2

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

  5. Taylor expanded around -inf 15.2

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  6. Simplified15.2

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2} + x\right)\right)}\]
  7. Using strategy rm

  8. Applied sin-sum0.4

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}\right)\]

  9. Final simplification0.4

    \[\leadsto \left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\]

Reproduce

herbie shell --seed 2019101 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))