Average Error: 39.7 → 0.4
Time: 21.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot -2
double f(double x, double eps) {
        double r553857 = x;
        double r553858 = eps;
        double r553859 = r553857 + r553858;
        double r553860 = cos(r553859);
        double r553861 = cos(r553857);
        double r553862 = r553860 - r553861;
        return r553862;
}
↓
double f(double x, double eps) {
        double r553863 = eps;
        double r553864 = 2.0;
        double r553865 = r553863 / r553864;
        double r553866 = sin(r553865);
        double r553867 = x;
        double r553868 = cos(r553867);
        double r553869 = 0.5;
        double r553870 = r553869 * r553863;
        double r553871 = sin(r553870);
        double r553872 = r553868 * r553871;
        double r553873 = sin(r553867);
        double r553874 = cos(r553870);
        double r553875 = r553873 * r553874;
        double r553876 = r553872 + r553875;
        double r553877 = r553866 * r553876;
        double r553878 = -2.0;
        double r553879 = r553877 * r553878;
        return r553879;
}

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-cos34.1

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

  4. Simplified14.9

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

  5. Taylor expanded around -inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  10. Taylor expanded around -inf 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))