Average Error: 39.4 → 0.5
Time: 22.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sqrt[3]{\cos x} \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) + \sin x \cdot \cos \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sqrt[3]{\cos x} \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) + \sin x \cdot \cos \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)
double f(double x, double eps) {
        double r1850046 = x;
        double r1850047 = eps;
        double r1850048 = r1850046 + r1850047;
        double r1850049 = cos(r1850048);
        double r1850050 = cos(r1850046);
        double r1850051 = r1850049 - r1850050;
        return r1850051;
}
↓
double f(double x, double eps) {
        double r1850052 = x;
        double r1850053 = cos(r1850052);
        double r1850054 = cbrt(r1850053);
        double r1850055 = eps;
        double r1850056 = 2.0;
        double r1850057 = r1850055 / r1850056;
        double r1850058 = sin(r1850057);
        double r1850059 = r1850054 * r1850054;
        double r1850060 = r1850058 * r1850059;
        double r1850061 = r1850054 * r1850060;
        double r1850062 = sin(r1850052);
        double r1850063 = cos(r1850057);
        double r1850064 = r1850062 * r1850063;
        double r1850065 = r1850061 + r1850064;
        double r1850066 = -2.0;
        double r1850067 = r1850058 * r1850066;
        double r1850068 = r1850065 * r1850067;
        return r1850068;
}

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 39.4

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

  3. Applied diff-cos33.2

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

  4. Simplified14.4

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

  5. Taylor expanded around inf 14.3

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

  6. Simplified14.3

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))