Average Error: 39.4 → 0.4
Time: 16.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right)
double f(double x, double eps) {
        double r34883 = x;
        double r34884 = eps;
        double r34885 = r34883 + r34884;
        double r34886 = cos(r34885);
        double r34887 = cos(r34883);
        double r34888 = r34886 - r34887;
        return r34888;
}
↓
double f(double x, double eps) {
        double r34889 = eps;
        double r34890 = 0.5;
        double r34891 = r34889 * r34890;
        double r34892 = sin(r34891);
        double r34893 = x;
        double r34894 = cos(r34893);
        double r34895 = sin(r34893);
        double r34896 = cos(r34891);
        double r34897 = r34895 * r34896;
        double r34898 = fma(r34894, r34892, r34897);
        double r34899 = -2.0;
        double r34900 = r34898 * r34899;
        double r34901 = r34892 * r34900;
        return r34901;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.4

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

  3. Applied diff-cos33.7

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

  4. Simplified14.9

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  10. Simplified0.4

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

  12. Applied fma-def0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))