Average Error: 39.9 → 0.4
Time: 20.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}{\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\frac{\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) - \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)}{\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r1026267 = x;
        double r1026268 = eps;
        double r1026269 = r1026267 + r1026268;
        double r1026270 = cos(r1026269);
        double r1026271 = cos(r1026267);
        double r1026272 = r1026270 - r1026271;
        return r1026272;
}
↓
double f(double x, double eps) {
        double r1026273 = x;
        double r1026274 = sin(r1026273);
        double r1026275 = 0.5;
        double r1026276 = eps;
        double r1026277 = r1026275 * r1026276;
        double r1026278 = cos(r1026277);
        double r1026279 = r1026274 * r1026278;
        double r1026280 = r1026279 * r1026279;
        double r1026281 = cos(r1026273);
        double r1026282 = sin(r1026277);
        double r1026283 = r1026281 * r1026282;
        double r1026284 = r1026283 * r1026283;
        double r1026285 = r1026280 - r1026284;
        double r1026286 = r1026279 - r1026283;
        double r1026287 = r1026285 / r1026286;
        double r1026288 = -2.0;
        double r1026289 = r1026282 * r1026288;
        double r1026290 = r1026287 * r1026289;
        return r1026290;
}

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 39.9

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

  3. Applied diff-cos34.2

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

  4. 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 flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))