Average Error: 39.3 → 0.4
Time: 34.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin x, \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin x, \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r33938 = x;
        double r33939 = eps;
        double r33940 = r33938 + r33939;
        double r33941 = cos(r33940);
        double r33942 = cos(r33938);
        double r33943 = r33941 - r33942;
        return r33943;
}
↓
double f(double x, double eps) {
        double r33944 = 0.5;
        double r33945 = eps;
        double r33946 = r33944 * r33945;
        double r33947 = cos(r33946);
        double r33948 = x;
        double r33949 = sin(r33948);
        double r33950 = sin(r33946);
        double r33951 = cos(r33948);
        double r33952 = r33950 * r33951;
        double r33953 = fma(r33947, r33949, r33952);
        double r33954 = expm1(r33953);
        double r33955 = log1p(r33954);
        double r33956 = -2.0;
        double r33957 = r33956 * r33950;
        double r33958 = r33955 * r33957;
        return r33958;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.3

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

  3. Applied diff-cos33.8

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.3

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

  11. Applied log1p-expm1-u0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))