Average Error: 39.5 → 0.5
Time: 23.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x + \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x + \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)
double f(double x, double eps) {
        double r2522615 = x;
        double r2522616 = eps;
        double r2522617 = r2522615 + r2522616;
        double r2522618 = cos(r2522617);
        double r2522619 = cos(r2522615);
        double r2522620 = r2522618 - r2522619;
        return r2522620;
}
↓
double f(double x, double eps) {
        double r2522621 = -2.0;
        double r2522622 = eps;
        double r2522623 = 0.5;
        double r2522624 = r2522622 * r2522623;
        double r2522625 = cos(r2522624);
        double r2522626 = x;
        double r2522627 = sin(r2522626);
        double r2522628 = r2522625 * r2522627;
        double r2522629 = sin(r2522624);
        double r2522630 = cos(r2522626);
        double r2522631 = cbrt(r2522630);
        double r2522632 = r2522631 * r2522631;
        double r2522633 = r2522629 * r2522632;
        double r2522634 = r2522633 * r2522631;
        double r2522635 = r2522628 + r2522634;
        double r2522636 = r2522635 * r2522629;
        double r2522637 = r2522621 * r2522636;
        return r2522637;
}

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 39.5

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

  3. Applied diff-cos34.2

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.3

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

  6. Simplified15.3

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

  8. Applied fma-udef15.3

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied associate-*r*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))