Average Error: 39.8 → 0.4
Time: 20.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\right)
double f(double x, double eps) {
        double r1613645 = x;
        double r1613646 = eps;
        double r1613647 = r1613645 + r1613646;
        double r1613648 = cos(r1613647);
        double r1613649 = cos(r1613645);
        double r1613650 = r1613648 - r1613649;
        return r1613650;
}
↓
double f(double x, double eps) {
        double r1613651 = 0.5;
        double r1613652 = eps;
        double r1613653 = r1613651 * r1613652;
        double r1613654 = sin(r1613653);
        double r1613655 = x;
        double r1613656 = cos(r1613655);
        double r1613657 = r1613656 * r1613654;
        double r1613658 = sin(r1613655);
        double r1613659 = cos(r1613653);
        double r1613660 = r1613658 * r1613659;
        double r1613661 = r1613657 + r1613660;
        double r1613662 = -2.0;
        double r1613663 = r1613661 * r1613662;
        double r1613664 = r1613654 * r1613663;
        return r1613664;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.8

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

  3. Applied diff-cos34.4

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

  4. Simplified15.4

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

  5. Taylor expanded around -inf 15.4

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

  6. Simplified15.4

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

  8. Applied fma-udef15.4

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))