Average Error: 39.1 → 0.5
Time: 34.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r53731 = x;
        double r53732 = eps;
        double r53733 = r53731 + r53732;
        double r53734 = cos(r53733);
        double r53735 = cos(r53731);
        double r53736 = r53734 - r53735;
        return r53736;
}
↓
double f(double x, double eps) {
        double r53737 = eps;
        double r53738 = 0.5;
        double r53739 = r53737 * r53738;
        double r53740 = sin(r53739);
        double r53741 = x;
        double r53742 = cos(r53741);
        double r53743 = r53740 * r53742;
        double r53744 = cos(r53739);
        double r53745 = cbrt(r53744);
        double r53746 = r53745 * r53745;
        double r53747 = sin(r53741);
        double r53748 = r53745 * r53747;
        double r53749 = r53746 * r53748;
        double r53750 = r53743 + r53749;
        double r53751 = -2.0;
        double r53752 = r53738 * r53737;
        double r53753 = sin(r53752);
        double r53754 = r53751 * r53753;
        double r53755 = r53750 * r53754;
        return r53755;
}

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 39.1

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

  3. Applied diff-cos33.5

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

    \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \color{blue}{\left(\left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right)} \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  12. Applied associate-*l*0.5

    \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \color{blue}{\left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x\right)}\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  13. Final simplification0.5

    \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))