Average Error: 39.7 → 0.4
Time: 19.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)
double f(double x, double eps) {
        double r1198908 = x;
        double r1198909 = eps;
        double r1198910 = r1198908 + r1198909;
        double r1198911 = cos(r1198910);
        double r1198912 = cos(r1198908);
        double r1198913 = r1198911 - r1198912;
        return r1198913;
}
↓
double f(double x, double eps) {
        double r1198914 = -2.0;
        double r1198915 = eps;
        double r1198916 = 2.0;
        double r1198917 = r1198915 / r1198916;
        double r1198918 = sin(r1198917);
        double r1198919 = x;
        double r1198920 = cos(r1198919);
        double r1198921 = 0.5;
        double r1198922 = r1198921 * r1198915;
        double r1198923 = sin(r1198922);
        double r1198924 = r1198920 * r1198923;
        double r1198925 = r1198924 * r1198924;
        double r1198926 = cos(r1198922);
        double r1198927 = sin(r1198919);
        double r1198928 = r1198926 * r1198927;
        double r1198929 = r1198928 * r1198928;
        double r1198930 = r1198925 - r1198929;
        double r1198931 = r1198924 - r1198928;
        double r1198932 = r1198930 / r1198931;
        double r1198933 = r1198918 * r1198932;
        double r1198934 = r1198914 * r1198933;
        return r1198934;
}

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 39.7

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

  3. Applied diff-cos34.1

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

  4. Simplified14.9

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

  5. Taylor expanded around -inf 14.9

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

  6. Simplified14.9

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))