Average Error: 39.2 → 0.4
Time: 26.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \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 x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)
double f(double x, double eps) {
        double r1928754 = x;
        double r1928755 = eps;
        double r1928756 = r1928754 + r1928755;
        double r1928757 = cos(r1928756);
        double r1928758 = cos(r1928754);
        double r1928759 = r1928757 - r1928758;
        return r1928759;
}
↓
double f(double x, double eps) {
        double r1928760 = -2.0;
        double r1928761 = x;
        double r1928762 = sin(r1928761);
        double r1928763 = r1928760 * r1928762;
        double r1928764 = eps;
        double r1928765 = 0.5;
        double r1928766 = r1928764 * r1928765;
        double r1928767 = cos(r1928766);
        double r1928768 = sin(r1928766);
        double r1928769 = r1928767 * r1928768;
        double r1928770 = r1928763 * r1928769;
        double r1928771 = r1928760 * r1928768;
        double r1928772 = cos(r1928761);
        double r1928773 = r1928772 * r1928768;
        double r1928774 = r1928771 * r1928773;
        double r1928775 = r1928770 + r1928774;
        return r1928775;
}

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 39.2

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

  3. Applied diff-cos33.7

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

  4. Simplified14.7

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

  5. Taylor expanded around inf 14.7

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

  6. Simplified14.7

    \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x + \varepsilon \cdot \frac{1}{2}\right)}\]
  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. Taylor expanded around inf 0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))