Average Error: 39.8 → 0.4
Time: 32.0s
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 r1740679 = x;
        double r1740680 = eps;
        double r1740681 = r1740679 + r1740680;
        double r1740682 = cos(r1740681);
        double r1740683 = cos(r1740679);
        double r1740684 = r1740682 - r1740683;
        return r1740684;
}
↓
double f(double x, double eps) {
        double r1740685 = x;
        double r1740686 = sin(r1740685);
        double r1740687 = 0.5;
        double r1740688 = eps;
        double r1740689 = r1740687 * r1740688;
        double r1740690 = cos(r1740689);
        double r1740691 = r1740686 * r1740690;
        double r1740692 = r1740691 * r1740691;
        double r1740693 = cos(r1740685);
        double r1740694 = sin(r1740689);
        double r1740695 = r1740693 * r1740694;
        double r1740696 = r1740695 * r1740695;
        double r1740697 = r1740692 - r1740696;
        double r1740698 = r1740691 - r1740695;
        double r1740699 = r1740697 / r1740698;
        double r1740700 = -2.0;
        double r1740701 = r1740694 * r1740700;
        double r1740702 = r1740699 * r1740701;
        return r1740702;
}

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 39.8

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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 2019151 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))