Average Error: 39.1 → 0.5
Time: 34.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin x\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \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) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r20452 = x;
        double r20453 = eps;
        double r20454 = r20452 + r20453;
        double r20455 = cos(r20454);
        double r20456 = cos(r20452);
        double r20457 = r20455 - r20456;
        return r20457;
}
↓
double f(double x, double eps) {
        double r20458 = eps;
        double r20459 = 0.5;
        double r20460 = r20458 * r20459;
        double r20461 = sin(r20460);
        double r20462 = x;
        double r20463 = cos(r20462);
        double r20464 = r20461 * r20463;
        double r20465 = cos(r20460);
        double r20466 = cbrt(r20465);
        double r20467 = r20466 * r20466;
        double r20468 = sin(r20462);
        double r20469 = r20466 * r20468;
        double r20470 = r20467 * r20469;
        double r20471 = r20464 + r20470;
        double r20472 = -2.0;
        double r20473 = r20459 * r20458;
        double r20474 = sin(r20473);
        double r20475 = r20472 * r20474;
        double r20476 = r20471 * r20475;
        return r20476;
}

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-cos33.5

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied associate-*l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))