Average Error: 40.4 → 0.5
Time: 23.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right)
double f(double x, double eps) {
        double r1848836 = x;
        double r1848837 = eps;
        double r1848838 = r1848836 + r1848837;
        double r1848839 = cos(r1848838);
        double r1848840 = cos(r1848836);
        double r1848841 = r1848839 - r1848840;
        return r1848841;
}
↓
double f(double x, double eps) {
        double r1848842 = eps;
        double r1848843 = 2.0;
        double r1848844 = r1848842 / r1848843;
        double r1848845 = sin(r1848844);
        double r1848846 = -2.0;
        double r1848847 = x;
        double r1848848 = cos(r1848847);
        double r1848849 = cbrt(r1848848);
        double r1848850 = r1848845 * r1848849;
        double r1848851 = r1848849 * r1848849;
        double r1848852 = r1848850 * r1848851;
        double r1848853 = cos(r1848844);
        double r1848854 = sin(r1848847);
        double r1848855 = r1848853 * r1848854;
        double r1848856 = r1848852 + r1848855;
        double r1848857 = r1848846 * r1848856;
        double r1848858 = r1848845 * r1848857;
        return r1848858;
}

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 40.4

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

  3. Applied diff-cos34.5

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

  4. Simplified15.6

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

  5. Taylor expanded around inf 15.6

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

  6. Simplified15.6

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

  8. Applied sin-sum0.4

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

  10. Applied add-cube-cbrt0.5

    \[\leadsto \left(-2 \cdot \left(\sin x \cdot \cos \left(\frac{\varepsilon}{2}\right) + \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

  11. Applied associate-*l*0.5

    \[\leadsto \left(-2 \cdot \left(\sin x \cdot \cos \left(\frac{\varepsilon}{2}\right) + \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

  12. Final simplification0.5

    \[\leadsto \sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right)\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))