Average Error: 40.0 → 0.4
Time: 1.4m
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r1674351 = x;
        double r1674352 = eps;
        double r1674353 = r1674351 + r1674352;
        double r1674354 = cos(r1674353);
        double r1674355 = cos(r1674351);
        double r1674356 = r1674354 - r1674355;
        return r1674356;
}
↓
double f(double x, double eps) {
        double r1674357 = 0.5;
        double r1674358 = eps;
        double r1674359 = r1674357 * r1674358;
        double r1674360 = sin(r1674359);
        double r1674361 = x;
        double r1674362 = cos(r1674361);
        double r1674363 = r1674360 * r1674362;
        double r1674364 = cos(r1674359);
        double r1674365 = sin(r1674361);
        double r1674366 = r1674364 * r1674365;
        double r1674367 = r1674363 + r1674366;
        double r1674368 = expm1(r1674367);
        double r1674369 = log1p(r1674368);
        double r1674370 = -2.0;
        double r1674371 = r1674360 * r1674370;
        double r1674372 = r1674369 * r1674371;
        return r1674372;
}

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 40.0

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

  3. Applied diff-cos34.3

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

  4. Simplified15.1

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

  6. Applied associate-*r*15.1

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

  8. Applied log1p-expm1-u15.1

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

  9. Simplified15.1

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

  11. Applied fma-udef15.1

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

  12. Applied sin-sum0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))