Average Error: 39.7 → 0.4
Time: 17.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) + \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) + \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right)
double f(double x, double eps) {
        double r1105849 = x;
        double r1105850 = eps;
        double r1105851 = r1105849 + r1105850;
        double r1105852 = cos(r1105851);
        double r1105853 = cos(r1105849);
        double r1105854 = r1105852 - r1105853;
        return r1105854;
}
↓
double f(double x, double eps) {
        double r1105855 = eps;
        double r1105856 = 0.5;
        double r1105857 = r1105855 * r1105856;
        double r1105858 = sin(r1105857);
        double r1105859 = x;
        double r1105860 = cos(r1105859);
        double r1105861 = r1105858 * r1105860;
        double r1105862 = -2.0;
        double r1105863 = r1105858 * r1105862;
        double r1105864 = r1105861 * r1105863;
        double r1105865 = sin(r1105859);
        double r1105866 = cos(r1105857);
        double r1105867 = r1105865 * r1105866;
        double r1105868 = r1105867 * r1105863;
        double r1105869 = r1105864 + r1105868;
        return r1105869;
}

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 39.7

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

  3. Applied diff-cos34.3

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

  4. Simplified15.3

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

  5. Taylor expanded around -inf 15.3

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

  6. Simplified15.3

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

  8. Applied sin-sum0.4

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

  10. Applied distribute-rgt-in0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))