Average Error: 39.6 → 0.4
Time: 24.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r1761227 = x;
        double r1761228 = eps;
        double r1761229 = r1761227 + r1761228;
        double r1761230 = cos(r1761229);
        double r1761231 = cos(r1761227);
        double r1761232 = r1761230 - r1761231;
        return r1761232;
}
↓
double f(double x, double eps) {
        double r1761233 = x;
        double r1761234 = cos(r1761233);
        double r1761235 = 0.5;
        double r1761236 = eps;
        double r1761237 = r1761235 * r1761236;
        double r1761238 = sin(r1761237);
        double r1761239 = r1761234 * r1761238;
        double r1761240 = sin(r1761233);
        double r1761241 = cos(r1761237);
        double r1761242 = r1761240 * r1761241;
        double r1761243 = r1761239 + r1761242;
        double r1761244 = -2.0;
        double r1761245 = r1761238 * r1761244;
        double r1761246 = r1761243 * r1761245;
        return r1761246;
}

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-cos34.3

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

  4. Simplified15.3

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

  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

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

  10. Applied +-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))