Average Error: 38.9 → 0.4
Time: 25.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)
double f(double x, double eps) {
        double r2870698 = x;
        double r2870699 = eps;
        double r2870700 = r2870698 + r2870699;
        double r2870701 = cos(r2870700);
        double r2870702 = cos(r2870698);
        double r2870703 = r2870701 - r2870702;
        return r2870703;
}
↓
double f(double x, double eps) {
        double r2870704 = -2.0;
        double r2870705 = eps;
        double r2870706 = 0.5;
        double r2870707 = r2870705 * r2870706;
        double r2870708 = sin(r2870707);
        double r2870709 = r2870704 * r2870708;
        double r2870710 = x;
        double r2870711 = cos(r2870710);
        double r2870712 = r2870711 * r2870708;
        double r2870713 = r2870709 * r2870712;
        double r2870714 = cos(r2870707);
        double r2870715 = sin(r2870710);
        double r2870716 = r2870714 * r2870715;
        double r2870717 = r2870709 * r2870716;
        double r2870718 = r2870713 + r2870717;
        return r2870718;
}

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 38.9

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

  3. Applied diff-cos33.4

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

  4. Simplified15.2

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

    \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x + \varepsilon \cdot \frac{1}{2}\right)}\]
  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. Applied distribute-rgt-in0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))