Average Error: 39.5 → 0.4
Time: 16.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r30780 = x;
        double r30781 = eps;
        double r30782 = r30780 + r30781;
        double r30783 = cos(r30782);
        double r30784 = cos(r30780);
        double r30785 = r30783 - r30784;
        return r30785;
}
↓
double f(double x, double eps) {
        double r30786 = 0.5;
        double r30787 = eps;
        double r30788 = r30786 * r30787;
        double r30789 = sin(r30788);
        double r30790 = x;
        double r30791 = cos(r30790);
        double r30792 = r30789 * r30791;
        double r30793 = -2.0;
        double r30794 = r30793 * r30789;
        double r30795 = r30792 * r30794;
        double r30796 = cos(r30788);
        double r30797 = sin(r30790);
        double r30798 = r30796 * r30797;
        double r30799 = r30798 * r30794;
        double r30800 = r30795 + r30799;
        return r30800;
}

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 39.5

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

  3. Applied diff-cos34.0

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

  4. Simplified14.5

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

  5. Taylor expanded around inf 14.5

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

  6. Simplified14.5

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

  8. Applied sin-sum0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))