Average Error: 39.8 → 1.0
Time: 22.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)} + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)} + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r3059977 = x;
        double r3059978 = eps;
        double r3059979 = r3059977 + r3059978;
        double r3059980 = cos(r3059979);
        double r3059981 = cos(r3059977);
        double r3059982 = r3059980 - r3059981;
        return r3059982;
}
↓
double f(double x, double eps) {
        double r3059983 = 0.5;
        double r3059984 = eps;
        double r3059985 = r3059983 * r3059984;
        double r3059986 = sin(r3059985);
        double r3059987 = -2.0;
        double r3059988 = x;
        double r3059989 = sin(r3059988);
        double r3059990 = cos(r3059985);
        double r3059991 = r3059989 * r3059990;
        double r3059992 = r3059991 * r3059991;
        double r3059993 = r3059991 * r3059992;
        double r3059994 = cbrt(r3059993);
        double r3059995 = cos(r3059988);
        double r3059996 = r3059995 * r3059986;
        double r3059997 = r3059994 + r3059996;
        double r3059998 = r3059987 * r3059997;
        double r3059999 = r3059986 * r3059998;
        return r3059999;
}

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 39.8

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.0

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

  8. Applied sin-sum0.4

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

  10. Applied add-cbrt-cube1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))