Average Error: 39.0 → 0.4
Time: 45.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2
double f(double x, double eps) {
        double r4296962 = x;
        double r4296963 = eps;
        double r4296964 = r4296962 + r4296963;
        double r4296965 = cos(r4296964);
        double r4296966 = cos(r4296962);
        double r4296967 = r4296965 - r4296966;
        return r4296967;
}
↓
double f(double x, double eps) {
        double r4296968 = x;
        double r4296969 = cos(r4296968);
        double r4296970 = 0.5;
        double r4296971 = eps;
        double r4296972 = r4296970 * r4296971;
        double r4296973 = sin(r4296972);
        double r4296974 = r4296969 * r4296973;
        double r4296975 = sin(r4296968);
        double r4296976 = cos(r4296972);
        double r4296977 = r4296975 * r4296976;
        double r4296978 = r4296974 + r4296977;
        double r4296979 = r4296978 * r4296973;
        double r4296980 = -2.0;
        double r4296981 = r4296979 * r4296980;
        return r4296981;
}

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 39.0

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

  3. Applied diff-cos33.4

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

  4. Simplified15.2

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

  6. Applied expm1-log1p-u15.3

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\right)\]

  7. Simplified15.3

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)\right)}\right)\right)\]
  8. Using strategy rm

  9. Applied fma-udef15.3

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + x\right)}\right)\right)\right)\]

  10. Applied sin-sum0.4

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)\right)\]
  11. Using strategy rm

  12. Applied expm1-log1p0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))