Average Error: 39.0 → 0.5
Time: 46.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\sin x \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \left(\sin x \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot -2
double f(double x, double eps) {
        double r1831998 = x;
        double r1831999 = eps;
        double r1832000 = r1831998 + r1831999;
        double r1832001 = cos(r1832000);
        double r1832002 = cos(r1831998);
        double r1832003 = r1832001 - r1832002;
        return r1832003;
}
↓
double f(double x, double eps) {
        double r1832004 = eps;
        double r1832005 = 0.5;
        double r1832006 = r1832004 * r1832005;
        double r1832007 = sin(r1832006);
        double r1832008 = x;
        double r1832009 = cos(r1832008);
        double r1832010 = r1832009 * r1832007;
        double r1832011 = sin(r1832008);
        double r1832012 = cos(r1832006);
        double r1832013 = cbrt(r1832012);
        double r1832014 = r1832013 * r1832013;
        double r1832015 = r1832011 * r1832014;
        double r1832016 = r1832015 * r1832013;
        double r1832017 = r1832010 + r1832016;
        double r1832018 = r1832007 * r1832017;
        double r1832019 = -2.0;
        double r1832020 = r1832018 * r1832019;
        return r1832020;
}

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 39.0

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

  3. Applied diff-cos33.4

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.2

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

  6. Simplified15.2

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

  8. Applied sin-sum0.4

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied associate-*r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))