Average Error: 39.7 → 0.5
Time: 23.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\left(\sin x \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\left(\sin x \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r2907623 = x;
        double r2907624 = eps;
        double r2907625 = r2907623 + r2907624;
        double r2907626 = cos(r2907625);
        double r2907627 = cos(r2907623);
        double r2907628 = r2907626 - r2907627;
        return r2907628;
}
↓
double f(double x, double eps) {
        double r2907629 = 0.5;
        double r2907630 = eps;
        double r2907631 = r2907629 * r2907630;
        double r2907632 = sin(r2907631);
        double r2907633 = -2.0;
        double r2907634 = x;
        double r2907635 = sin(r2907634);
        double r2907636 = cos(r2907631);
        double r2907637 = cbrt(r2907636);
        double r2907638 = r2907635 * r2907637;
        double r2907639 = r2907637 * r2907637;
        double r2907640 = r2907638 * r2907639;
        double r2907641 = cos(r2907634);
        double r2907642 = r2907641 * r2907632;
        double r2907643 = r2907640 + r2907642;
        double r2907644 = r2907633 * r2907643;
        double r2907645 = r2907632 * r2907644;
        return r2907645;
}

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 39.7

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

    \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2} + x\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\]
  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-cube-cbrt0.5

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

  11. Applied associate-*l*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))