Average Error: 39.9 → 0.5
Time: 23.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)
double f(double x, double eps) {
        double r1702867 = x;
        double r1702868 = eps;
        double r1702869 = r1702867 + r1702868;
        double r1702870 = cos(r1702869);
        double r1702871 = cos(r1702867);
        double r1702872 = r1702870 - r1702871;
        return r1702872;
}
↓
double f(double x, double eps) {
        double r1702873 = -2.0;
        double r1702874 = 0.5;
        double r1702875 = eps;
        double r1702876 = r1702874 * r1702875;
        double r1702877 = cos(r1702876);
        double r1702878 = cbrt(r1702877);
        double r1702879 = x;
        double r1702880 = sin(r1702879);
        double r1702881 = r1702878 * r1702880;
        double r1702882 = r1702878 * r1702878;
        double r1702883 = r1702881 * r1702882;
        double r1702884 = sin(r1702876);
        double r1702885 = cos(r1702879);
        double r1702886 = r1702884 * r1702885;
        double r1702887 = r1702883 + r1702886;
        double r1702888 = r1702873 * r1702887;
        double r1702889 = r1702888 * r1702884;
        return r1702889;
}

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

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

  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

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

  8. Applied fma-udef15.4

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

    \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right)} \cdot \sin x\right) \cdot -2\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\]

  12. Applied associate-*l*0.5

    \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)}\right) \cdot -2\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\]

  13. Final simplification0.5

    \[\leadsto \left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\]

Reproduce

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