Average Error: 39.5 → 0.4
Time: 26.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\cos x \cdot \left(\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\cos x \cdot \left(\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) + \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)
double f(double x, double eps) {
        double r1432197 = x;
        double r1432198 = eps;
        double r1432199 = r1432197 + r1432198;
        double r1432200 = cos(r1432199);
        double r1432201 = cos(r1432197);
        double r1432202 = r1432200 - r1432201;
        return r1432202;
}
↓
double f(double x, double eps) {
        double r1432203 = x;
        double r1432204 = cos(r1432203);
        double r1432205 = -2.0;
        double r1432206 = eps;
        double r1432207 = 0.5;
        double r1432208 = r1432206 * r1432207;
        double r1432209 = sin(r1432208);
        double r1432210 = r1432205 * r1432209;
        double r1432211 = r1432210 * r1432209;
        double r1432212 = r1432204 * r1432211;
        double r1432213 = cos(r1432208);
        double r1432214 = sin(r1432203);
        double r1432215 = r1432213 * r1432214;
        double r1432216 = r1432210 * r1432215;
        double r1432217 = r1432212 + r1432216;
        return r1432217;
}

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 39.5

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

  3. Applied diff-cos33.9

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

  4. Simplified15.1

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

  5. Taylor expanded around -inf 15.0

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

  6. Simplified15.0

    \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x + \varepsilon \cdot \frac{1}{2}\right)}\]
  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. Applied distribute-rgt-in0.4

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

  11. Applied associate-*l*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))