Average Error: 39.8 → 1.0
Time: 23.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)} + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \left(\sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)} + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r2815821 = x;
        double r2815822 = eps;
        double r2815823 = r2815821 + r2815822;
        double r2815824 = cos(r2815823);
        double r2815825 = cos(r2815821);
        double r2815826 = r2815824 - r2815825;
        return r2815826;
}
↓
double f(double x, double eps) {
        double r2815827 = 0.5;
        double r2815828 = eps;
        double r2815829 = r2815827 * r2815828;
        double r2815830 = sin(r2815829);
        double r2815831 = -2.0;
        double r2815832 = x;
        double r2815833 = sin(r2815832);
        double r2815834 = cos(r2815829);
        double r2815835 = r2815833 * r2815834;
        double r2815836 = r2815835 * r2815835;
        double r2815837 = r2815835 * r2815836;
        double r2815838 = cbrt(r2815837);
        double r2815839 = cos(r2815832);
        double r2815840 = r2815839 * r2815830;
        double r2815841 = r2815838 + r2815840;
        double r2815842 = r2815831 * r2815841;
        double r2815843 = r2815830 * r2815842;
        return r2815843;
}

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 39.8

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

  3. Applied diff-cos34.1

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.0

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

  8. Applied sin-sum0.4

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

  10. Applied add-cbrt-cube1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))