Average Error: 39.6 → 0.5
Time: 25.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x}\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x}\right)\right)
double f(double x, double eps) {
        double r1392662 = x;
        double r1392663 = eps;
        double r1392664 = r1392662 + r1392663;
        double r1392665 = cos(r1392664);
        double r1392666 = cos(r1392662);
        double r1392667 = r1392665 - r1392666;
        return r1392667;
}
↓
double f(double x, double eps) {
        double r1392668 = 0.5;
        double r1392669 = eps;
        double r1392670 = r1392668 * r1392669;
        double r1392671 = sin(r1392670);
        double r1392672 = -2.0;
        double r1392673 = cos(r1392670);
        double r1392674 = x;
        double r1392675 = sin(r1392674);
        double r1392676 = r1392673 * r1392675;
        double r1392677 = cos(r1392674);
        double r1392678 = cbrt(r1392677);
        double r1392679 = r1392678 * r1392678;
        double r1392680 = r1392671 * r1392679;
        double r1392681 = r1392680 * r1392678;
        double r1392682 = r1392676 + r1392681;
        double r1392683 = r1392672 * r1392682;
        double r1392684 = r1392671 * r1392683;
        return r1392684;
}

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 39.6

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

  3. Applied diff-cos34.2

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

  4. Simplified15.3

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

  5. Taylor expanded around -inf 15.3

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

  6. Simplified15.3

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

  8. Applied fma-udef15.3

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied associate-*r*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))