Average Error: 39.1 → 1.8
Time: 38.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \left(\sqrt[3]{\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)} + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \left(\sqrt[3]{\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)} + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)
double f(double x, double eps) {
        double r3431421 = x;
        double r3431422 = eps;
        double r3431423 = r3431421 + r3431422;
        double r3431424 = cos(r3431423);
        double r3431425 = cos(r3431421);
        double r3431426 = r3431424 - r3431425;
        return r3431426;
}
↓
double f(double x, double eps) {
        double r3431427 = -2.0;
        double r3431428 = x;
        double r3431429 = cos(r3431428);
        double r3431430 = eps;
        double r3431431 = 2.0;
        double r3431432 = r3431430 / r3431431;
        double r3431433 = sin(r3431432);
        double r3431434 = r3431429 * r3431433;
        double r3431435 = r3431434 * r3431434;
        double r3431436 = r3431434 * r3431435;
        double r3431437 = cbrt(r3431436);
        double r3431438 = cos(r3431432);
        double r3431439 = sin(r3431428);
        double r3431440 = r3431438 * r3431439;
        double r3431441 = r3431437 + r3431440;
        double r3431442 = r3431427 * r3431441;
        double r3431443 = r3431442 * r3431433;
        return r3431443;
}

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 39.1

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

  3. Applied diff-cos33.8

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

  4. Simplified15.0

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

  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. Simplified14.9

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

  8. Applied sin-sum0.4

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

  10. Applied add-cbrt-cube1.7

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

  11. Applied add-cbrt-cube1.8

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

  12. Applied cbrt-unprod1.8

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

  13. Simplified1.8

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

  14. Final simplification1.8

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))