Average Error: 39.9 → 0.4
Time: 22.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)
double f(double x, double eps) {
        double r790607 = x;
        double r790608 = eps;
        double r790609 = r790607 + r790608;
        double r790610 = cos(r790609);
        double r790611 = cos(r790607);
        double r790612 = r790610 - r790611;
        return r790612;
}
↓
double f(double x, double eps) {
        double r790613 = eps;
        double r790614 = 2.0;
        double r790615 = r790613 / r790614;
        double r790616 = sin(r790615);
        double r790617 = x;
        double r790618 = cos(r790617);
        double r790619 = r790616 * r790618;
        double r790620 = cos(r790615);
        double r790621 = sin(r790617);
        double r790622 = r790620 * r790621;
        double r790623 = r790619 + r790622;
        double r790624 = -2.0;
        double r790625 = r790616 * r790624;
        double r790626 = r790623 * r790625;
        return r790626;
}

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 39.9

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

  3. Applied diff-cos34.1

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

  4. Simplified15.4

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

  5. Taylor expanded around inf 15.4

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

  6. Simplified15.4

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

  8. Applied fma-udef15.4

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

  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

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))