Average Error: 39.3 → 0.4
Time: 24.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \mathsf{fma}\left(\sin x, \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \mathsf{fma}\left(\sin x, \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)
double f(double x, double eps) {
        double r1653750 = x;
        double r1653751 = eps;
        double r1653752 = r1653750 + r1653751;
        double r1653753 = cos(r1653752);
        double r1653754 = cos(r1653750);
        double r1653755 = r1653753 - r1653754;
        return r1653755;
}
↓
double f(double x, double eps) {
        double r1653756 = 0.5;
        double r1653757 = eps;
        double r1653758 = r1653756 * r1653757;
        double r1653759 = sin(r1653758);
        double r1653760 = -2.0;
        double r1653761 = r1653759 * r1653760;
        double r1653762 = x;
        double r1653763 = sin(r1653762);
        double r1653764 = cos(r1653758);
        double r1653765 = cos(r1653762);
        double r1653766 = r1653759 * r1653765;
        double r1653767 = fma(r1653763, r1653764, r1653766);
        double r1653768 = r1653761 * r1653767;
        return r1653768;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.3

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

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

  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

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  10. Taylor expanded around inf 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))