Average Error: 39.1 → 0.5
Time: 30.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r59255 = x;
        double r59256 = eps;
        double r59257 = r59255 + r59256;
        double r59258 = cos(r59257);
        double r59259 = cos(r59255);
        double r59260 = r59258 - r59259;
        return r59260;
}
↓
double f(double x, double eps) {
        double r59261 = 0.5;
        double r59262 = eps;
        double r59263 = r59261 * r59262;
        double r59264 = sin(r59263);
        double r59265 = x;
        double r59266 = cos(r59265);
        double r59267 = r59264 * r59266;
        double r59268 = cos(r59263);
        double r59269 = cbrt(r59268);
        double r59270 = r59269 * r59269;
        double r59271 = sin(r59265);
        double r59272 = r59269 * r59271;
        double r59273 = r59270 * r59272;
        double r59274 = r59267 + r59273;
        double r59275 = -2.0;
        double r59276 = r59264 * r59275;
        double r59277 = r59274 * r59276;
        return r59277;
}

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

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied sin-sum0.4

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*l*0.5

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

  12. Final simplification0.5

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

Reproduce

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