Average Error: 40.4 → 0.4
Time: 21.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin x, \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\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 r1691042 = x;
        double r1691043 = eps;
        double r1691044 = r1691042 + r1691043;
        double r1691045 = cos(r1691044);
        double r1691046 = cos(r1691042);
        double r1691047 = r1691045 - r1691046;
        return r1691047;
}
↓
double f(double x, double eps) {
        double r1691048 = 0.5;
        double r1691049 = eps;
        double r1691050 = r1691048 * r1691049;
        double r1691051 = sin(r1691050);
        double r1691052 = -2.0;
        double r1691053 = r1691051 * r1691052;
        double r1691054 = cos(r1691050);
        double r1691055 = x;
        double r1691056 = sin(r1691055);
        double r1691057 = cos(r1691055);
        double r1691058 = r1691051 * r1691057;
        double r1691059 = fma(r1691054, r1691056, r1691058);
        double r1691060 = r1691053 * r1691059;
        return r1691060;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 40.4

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

  3. Applied diff-cos34.5

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

  4. Simplified15.6

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

  5. Taylor expanded around inf 15.6

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

  6. Simplified15.6

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

  8. Applied fma-udef15.6

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

  9. Applied sin-sum0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Taylor expanded around inf 0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))