Average Error: 39.6 → 0.4
Time: 23.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right)\]
double f(double x, double eps) {
        double r2299737 = x;
        double r2299738 = eps;
        double r2299739 = r2299737 + r2299738;
        double r2299740 = cos(r2299739);
        double r2299741 = cos(r2299737);
        double r2299742 = r2299740 - r2299741;
        return r2299742;
}
↓
double f(double x, double eps) {
        double r2299743 = eps;
        double r2299744 = 2.0;
        double r2299745 = r2299743 / r2299744;
        double r2299746 = sin(r2299745);
        double r2299747 = x;
        double r2299748 = cos(r2299747);
        double r2299749 = 0.5;
        double r2299750 = r2299743 * r2299749;
        double r2299751 = sin(r2299750);
        double r2299752 = r2299748 * r2299751;
        double r2299753 = sin(r2299747);
        double r2299754 = cos(r2299750);
        double r2299755 = r2299753 * r2299754;
        double r2299756 = r2299752 + r2299755;
        double r2299757 = -2.0;
        double r2299758 = r2299756 * r2299757;
        double r2299759 = r2299746 * r2299758;
        return r2299759;
}


\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right)

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.6

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

  3. Applied diff-cos33.8

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

  4. Simplified15.1

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

  6. Applied associate-*r*15.1

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

  7. Taylor expanded around inf 15.1

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

  8. Simplified15.1

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

  10. Applied fma-udef15.1

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

  11. Applied sin-sum0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))