Average Error: 39.9 → 0.4
Time: 20.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin x \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\right) \cdot \cos \left(\frac{\varepsilon}{2}\right) + \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin x \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\right) \cdot \cos \left(\frac{\varepsilon}{2}\right) + \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)
double f(double x, double eps) {
        double r1933688 = x;
        double r1933689 = eps;
        double r1933690 = r1933688 + r1933689;
        double r1933691 = cos(r1933690);
        double r1933692 = cos(r1933688);
        double r1933693 = r1933691 - r1933692;
        return r1933693;
}
↓
double f(double x, double eps) {
        double r1933694 = x;
        double r1933695 = sin(r1933694);
        double r1933696 = eps;
        double r1933697 = 2.0;
        double r1933698 = r1933696 / r1933697;
        double r1933699 = sin(r1933698);
        double r1933700 = -2.0;
        double r1933701 = r1933699 * r1933700;
        double r1933702 = r1933695 * r1933701;
        double r1933703 = cos(r1933698);
        double r1933704 = r1933702 * r1933703;
        double r1933705 = cos(r1933694);
        double r1933706 = r1933699 * r1933705;
        double r1933707 = r1933706 * r1933701;
        double r1933708 = r1933704 + r1933707;
        return r1933708;
}

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. Using strategy rm

  3. Applied diff-cos34.1

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

  4. Simplified15.4

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

  5. Taylor expanded around inf 15.4

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

  6. Simplified15.4

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

  8. Applied sin-sum0.4

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

  9. Applied distribute-rgt-in0.4

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

  11. Applied associate-*l*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))