Average Error: 0.0 → 0.0
Time: 24.8s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\sqrt[3]{(\left(\frac{1}{(x \cdot x + -1)_*}\right) \cdot \left(x + 1\right) + \left(\frac{x}{x + 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right)}\]
double f(double x) {
        double r15682783 = 1.0;
        double r15682784 = x;
        double r15682785 = r15682784 - r15682783;
        double r15682786 = r15682783 / r15682785;
        double r15682787 = r15682784 + r15682783;
        double r15682788 = r15682784 / r15682787;
        double r15682789 = r15682786 + r15682788;
        return r15682789;
}
↓
double f(double x) {
        double r15682790 = 1.0;
        double r15682791 = x;
        double r15682792 = -1.0;
        double r15682793 = fma(r15682791, r15682791, r15682792);
        double r15682794 = r15682790 / r15682793;
        double r15682795 = r15682791 + r15682790;
        double r15682796 = r15682791 / r15682795;
        double r15682797 = fma(r15682794, r15682795, r15682796);
        double r15682798 = r15682791 - r15682790;
        double r15682799 = r15682790 / r15682798;
        double r15682800 = r15682799 + r15682796;
        double r15682801 = r15682800 * r15682800;
        double r15682802 = r15682797 * r15682801;
        double r15682803 = cbrt(r15682802);
        return r15682803;
}


\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\sqrt[3]{(\left(\frac{1}{(x \cdot x + -1)_*}\right) \cdot \left(x + 1\right) + \left(\frac{x}{x + 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right)}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}}\]
  4. Using strategy rm

  5. Applied flip--0.0

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

  6. Applied associate-/r/0.0

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

  7. Applied fma-def0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \color{blue}{(\left(\frac{1}{x \cdot x - 1 \cdot 1}\right) \cdot \left(x + 1\right) + \left(\frac{x}{x + 1}\right))_*}}\]

  8. Simplified0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot (\color{blue}{\left(\frac{1}{(x \cdot x + -1)_*}\right)} \cdot \left(x + 1\right) + \left(\frac{x}{x + 1}\right))_*}\]

  9. Final simplification0.0

    \[\leadsto \sqrt[3]{(\left(\frac{1}{(x \cdot x + -1)_*}\right) \cdot \left(x + 1\right) + \left(\frac{x}{x + 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right)}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))