Average Error: 40.0 → 0.4
Time: 19.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)
double f(double x, double eps) {
        double r736398 = x;
        double r736399 = eps;
        double r736400 = r736398 + r736399;
        double r736401 = cos(r736400);
        double r736402 = cos(r736398);
        double r736403 = r736401 - r736402;
        return r736403;
}
↓
double f(double x, double eps) {
        double r736404 = -2.0;
        double r736405 = x;
        double r736406 = r736405 + r736405;
        double r736407 = 0.5;
        double r736408 = r736406 * r736407;
        double r736409 = cos(r736408);
        double r736410 = eps;
        double r736411 = r736410 * r736407;
        double r736412 = sin(r736411);
        double r736413 = r736409 * r736412;
        double r736414 = r736404 * r736413;
        double r736415 = r736414 * r736412;
        double r736416 = sin(r736408);
        double r736417 = cos(r736411);
        double r736418 = r736416 * r736417;
        double r736419 = r736418 * r736404;
        double r736420 = r736419 * r736412;
        double r736421 = r736415 + r736420;
        return r736421;
}

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 40.0

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.4

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

  5. Taylor expanded around -inf 15.4

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

  6. Simplified15.4

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

  8. Applied distribute-rgt-in15.4

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

  9. Applied sin-sum0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Applied distribute-lft-in0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))