Average Error: 39.4 → 0.5
Time: 16.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r20573 = x;
        double r20574 = eps;
        double r20575 = r20573 + r20574;
        double r20576 = cos(r20575);
        double r20577 = cos(r20573);
        double r20578 = r20576 - r20577;
        return r20578;
}
↓
double f(double x, double eps) {
        double r20579 = -2.0;
        double r20580 = 0.5;
        double r20581 = eps;
        double r20582 = r20580 * r20581;
        double r20583 = sin(r20582);
        double r20584 = r20579 * r20583;
        double r20585 = x;
        double r20586 = cos(r20585);
        double r20587 = cbrt(r20586);
        double r20588 = r20587 * r20587;
        double r20589 = r20587 * r20583;
        double r20590 = r20588 * r20589;
        double r20591 = sin(r20585);
        double r20592 = cos(r20582);
        double r20593 = r20591 * r20592;
        double r20594 = r20590 + r20593;
        double r20595 = r20584 * r20594;
        return r20595;
}

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 39.4

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

  3. Applied diff-cos33.7

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

  4. Simplified14.7

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

  5. Taylor expanded around inf 14.7

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

  6. Simplified14.7

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

  8. Applied sin-sum0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-cube-cbrt0.5

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

  13. Applied associate-*l*0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019305 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))