Average Error: 39.1 → 0.4
Time: 40.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)
double f(double x, double eps) {
        double r3219976 = x;
        double r3219977 = eps;
        double r3219978 = r3219976 + r3219977;
        double r3219979 = cos(r3219978);
        double r3219980 = cos(r3219976);
        double r3219981 = r3219979 - r3219980;
        return r3219981;
}
↓
double f(double x, double eps) {
        double r3219982 = eps;
        double r3219983 = 0.5;
        double r3219984 = r3219982 * r3219983;
        double r3219985 = sin(r3219984);
        double r3219986 = -2.0;
        double r3219987 = r3219985 * r3219986;
        double r3219988 = cos(r3219984);
        double r3219989 = x;
        double r3219990 = sin(r3219989);
        double r3219991 = cos(r3219989);
        double r3219992 = r3219985 * r3219991;
        double r3219993 = fma(r3219988, r3219990, r3219992);
        double r3219994 = r3219987 * r3219993;
        return r3219994;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.1

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

  3. Applied diff-cos33.8

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  10. Taylor expanded around inf 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))