Average Error: 39.6 → 0.4
Time: 23.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot -2\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\mathsf{log1p}\left(\mathsf{expm1}\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\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)
double f(double x, double eps) {
        double r1022511 = x;
        double r1022512 = eps;
        double r1022513 = r1022511 + r1022512;
        double r1022514 = cos(r1022513);
        double r1022515 = cos(r1022511);
        double r1022516 = r1022514 - r1022515;
        return r1022516;
}
↓
double f(double x, double eps) {
        double r1022517 = x;
        double r1022518 = cos(r1022517);
        double r1022519 = 0.5;
        double r1022520 = eps;
        double r1022521 = r1022519 * r1022520;
        double r1022522 = sin(r1022521);
        double r1022523 = r1022518 * r1022522;
        double r1022524 = sin(r1022517);
        double r1022525 = cos(r1022521);
        double r1022526 = r1022524 * r1022525;
        double r1022527 = r1022523 + r1022526;
        double r1022528 = expm1(r1022527);
        double r1022529 = log1p(r1022528);
        double r1022530 = -2.0;
        double r1022531 = r1022529 * r1022530;
        double r1022532 = r1022531 * r1022522;
        return r1022532;
}

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 39.6

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.3

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

  5. Taylor expanded around inf 15.3

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

  6. Simplified15.3

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

  8. Applied fma-udef15.3

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

  9. Applied sin-sum0.4

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

  11. Applied log1p-expm1-u0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))