Average Error: 40.4 → 0.5
Time: 21.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) + \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)\right)
double f(double x, double eps) {
        double r1940362 = x;
        double r1940363 = eps;
        double r1940364 = r1940362 + r1940363;
        double r1940365 = cos(r1940364);
        double r1940366 = cos(r1940362);
        double r1940367 = r1940365 - r1940366;
        return r1940367;
}
↓
double f(double x, double eps) {
        double r1940368 = eps;
        double r1940369 = 2.0;
        double r1940370 = r1940368 / r1940369;
        double r1940371 = sin(r1940370);
        double r1940372 = -2.0;
        double r1940373 = x;
        double r1940374 = cos(r1940373);
        double r1940375 = cbrt(r1940374);
        double r1940376 = r1940371 * r1940375;
        double r1940377 = r1940375 * r1940375;
        double r1940378 = r1940376 * r1940377;
        double r1940379 = cos(r1940370);
        double r1940380 = sin(r1940373);
        double r1940381 = r1940379 * r1940380;
        double r1940382 = r1940378 + r1940381;
        double r1940383 = r1940372 * r1940382;
        double r1940384 = r1940371 * r1940383;
        return r1940384;
}

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 40.4

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

  3. Applied diff-cos34.5

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

  4. Simplified15.6

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

  5. Taylor expanded around inf 15.6

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

  6. Simplified15.6

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

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

  11. Applied associate-*l*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))