Average Error: 39.1 → 0.5
Time: 34.6s
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 r20415 = x;
        double r20416 = eps;
        double r20417 = r20415 + r20416;
        double r20418 = cos(r20417);
        double r20419 = cos(r20415);
        double r20420 = r20418 - r20419;
        return r20420;
}
↓
double f(double x, double eps) {
        double r20421 = eps;
        double r20422 = 0.5;
        double r20423 = r20421 * r20422;
        double r20424 = sin(r20423);
        double r20425 = x;
        double r20426 = cos(r20425);
        double r20427 = r20424 * r20426;
        double r20428 = cos(r20423);
        double r20429 = cbrt(r20428);
        double r20430 = r20429 * r20429;
        double r20431 = sin(r20425);
        double r20432 = r20429 * r20431;
        double r20433 = r20430 * r20432;
        double r20434 = r20427 + r20433;
        double r20435 = -2.0;
        double r20436 = r20422 * r20421;
        double r20437 = sin(r20436);
        double r20438 = r20435 * r20437;
        double r20439 = r20434 * r20438;
        return r20439;
}

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)))