Average Error: 39.6 → 0.4
Time: 25.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}{\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\frac{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}{\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r1616409 = x;
        double r1616410 = eps;
        double r1616411 = r1616409 + r1616410;
        double r1616412 = cos(r1616411);
        double r1616413 = cos(r1616409);
        double r1616414 = r1616412 - r1616413;
        return r1616414;
}
↓
double f(double x, double eps) {
        double r1616415 = x;
        double r1616416 = sin(r1616415);
        double r1616417 = 0.5;
        double r1616418 = eps;
        double r1616419 = r1616417 * r1616418;
        double r1616420 = cos(r1616419);
        double r1616421 = r1616416 * r1616420;
        double r1616422 = r1616421 * r1616421;
        double r1616423 = cos(r1616415);
        double r1616424 = sin(r1616419);
        double r1616425 = r1616423 * r1616424;
        double r1616426 = r1616425 * r1616425;
        double r1616427 = r1616422 - r1616426;
        double r1616428 = r1616421 - r1616425;
        double r1616429 = r1616427 / r1616428;
        double r1616430 = -2.0;
        double r1616431 = r1616424 * r1616430;
        double r1616432 = r1616429 * r1616431;
        return r1616432;
}

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-cos34.2

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

  4. Simplified15.3

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

  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

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))