Average Error: 39.7 → 0.4
Time: 26.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin x \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot -2 + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin x \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right) \cdot -2 + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r3615645 = x;
        double r3615646 = eps;
        double r3615647 = r3615645 + r3615646;
        double r3615648 = cos(r3615647);
        double r3615649 = cos(r3615645);
        double r3615650 = r3615648 - r3615649;
        return r3615650;
}
↓
double f(double x, double eps) {
        double r3615651 = x;
        double r3615652 = sin(r3615651);
        double r3615653 = 0.5;
        double r3615654 = eps;
        double r3615655 = r3615653 * r3615654;
        double r3615656 = sin(r3615655);
        double r3615657 = cos(r3615655);
        double r3615658 = r3615656 * r3615657;
        double r3615659 = r3615652 * r3615658;
        double r3615660 = -2.0;
        double r3615661 = r3615659 * r3615660;
        double r3615662 = r3615660 * r3615656;
        double r3615663 = cos(r3615651);
        double r3615664 = r3615663 * r3615656;
        double r3615665 = r3615662 * r3615664;
        double r3615666 = r3615661 + r3615665;
        return r3615666;
}

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 39.7

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.1

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

  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-lft-in0.4

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

  10. Taylor expanded around -inf 0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))