Average Error: 39.6 → 0.4
Time: 31.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)
double f(double x, double eps) {
        double r39933 = x;
        double r39934 = eps;
        double r39935 = r39933 + r39934;
        double r39936 = cos(r39935);
        double r39937 = cos(r39933);
        double r39938 = r39936 - r39937;
        return r39938;
}
↓
double f(double x, double eps) {
        double r39939 = -2.0;
        double r39940 = 0.5;
        double r39941 = eps;
        double r39942 = r39940 * r39941;
        double r39943 = sin(r39942);
        double r39944 = x;
        double r39945 = cos(r39944);
        double r39946 = cos(r39942);
        double r39947 = sin(r39944);
        double r39948 = r39946 * r39947;
        double r39949 = fma(r39943, r39945, r39948);
        double r39950 = r39943 * r39949;
        double r39951 = r39939 * r39950;
        return r39951;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.6

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

  3. Applied diff-cos34.0

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

  4. Simplified15.1

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

  6. Applied expm1-log1p-u15.1

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

  7. Simplified15.1

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

  9. Applied fma-udef15.1

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

  10. Applied sin-sum0.4

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

  11. Taylor expanded around inf 0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))