Average Error: 39.7 → 0.3
Time: 20.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin x, \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\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)
double f(double x, double eps) {
        double r46569 = x;
        double r46570 = eps;
        double r46571 = r46569 + r46570;
        double r46572 = cos(r46571);
        double r46573 = cos(r46569);
        double r46574 = r46572 - r46573;
        return r46574;
}
↓
double f(double x, double eps) {
        double r46575 = -2.0;
        double r46576 = 0.5;
        double r46577 = eps;
        double r46578 = r46576 * r46577;
        double r46579 = sin(r46578);
        double r46580 = r46575 * r46579;
        double r46581 = cos(r46578);
        double r46582 = x;
        double r46583 = sin(r46582);
        double r46584 = cos(r46582);
        double r46585 = r46579 * r46584;
        double r46586 = fma(r46581, r46583, r46585);
        double r46587 = r46580 * r46586;
        return r46587;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.7

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied associate-*l*0.4

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))