Average Error: 39.8 → 0.4
Time: 17.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r20828 = x;
        double r20829 = eps;
        double r20830 = r20828 + r20829;
        double r20831 = cos(r20830);
        double r20832 = cos(r20828);
        double r20833 = r20831 - r20832;
        return r20833;
}
↓
double f(double x, double eps) {
        double r20834 = 0.5;
        double r20835 = eps;
        double r20836 = r20834 * r20835;
        double r20837 = sin(r20836);
        double r20838 = x;
        double r20839 = cos(r20838);
        double r20840 = cos(r20836);
        double r20841 = sin(r20838);
        double r20842 = r20840 * r20841;
        double r20843 = fma(r20837, r20839, r20842);
        double r20844 = -2.0;
        double r20845 = r20837 * r20844;
        double r20846 = r20843 * r20845;
        return r20846;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.8

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

  3. Applied diff-cos34.4

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied fma-udef15.1

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

  9. Applied sin-sum0.4

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

  11. Applied fma-def0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))