Average Error: 39.9 → 0.4
Time: 16.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)
double f(double x, double eps) {
        double r34698 = x;
        double r34699 = eps;
        double r34700 = r34698 + r34699;
        double r34701 = cos(r34700);
        double r34702 = cos(r34698);
        double r34703 = r34701 - r34702;
        return r34703;
}
↓
double f(double x, double eps) {
        double r34704 = -2.0;
        double r34705 = 0.5;
        double r34706 = eps;
        double r34707 = r34705 * r34706;
        double r34708 = cos(r34707);
        double r34709 = x;
        double r34710 = sin(r34709);
        double r34711 = sin(r34707);
        double r34712 = r34710 * r34711;
        double r34713 = r34708 * r34712;
        double r34714 = r34704 * r34713;
        double r34715 = r34711 * r34704;
        double r34716 = cos(r34709);
        double r34717 = r34711 * r34716;
        double r34718 = r34715 * r34717;
        double r34719 = r34714 + r34718;
        return r34719;
}

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 39.9

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

  2. Simplified39.9

    \[\leadsto \color{blue}{\cos \left(\varepsilon + x\right) - \cos x}\]
  3. Using strategy rm

  4. Applied diff-cos34.3

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

  5. Simplified15.6

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

  6. Taylor expanded around inf 15.6

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

  7. Simplified15.6

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

  9. Applied sin-sum0.4

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

  10. Applied distribute-lft-in0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Simplified0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))