Average Error: 39.5 → 0.4
Time: 16.5s
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 r31478 = x;
        double r31479 = eps;
        double r31480 = r31478 + r31479;
        double r31481 = cos(r31480);
        double r31482 = cos(r31478);
        double r31483 = r31481 - r31482;
        return r31483;
}
↓
double f(double x, double eps) {
        double r31484 = eps;
        double r31485 = 0.5;
        double r31486 = r31484 * r31485;
        double r31487 = sin(r31486);
        double r31488 = x;
        double r31489 = cos(r31488);
        double r31490 = sin(r31488);
        double r31491 = cos(r31486);
        double r31492 = r31490 * r31491;
        double r31493 = fma(r31489, r31487, r31492);
        double r31494 = -2.0;
        double r31495 = r31493 * r31494;
        double r31496 = r31487 * r31495;
        return r31496;
}

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-cos33.9

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

  4. Simplified15.2

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

  5. Taylor expanded around inf 15.2

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

  6. Simplified15.2

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

  8. Applied fma-udef15.2

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

  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 2019194 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))