Average Error: 0.0 → 0.0
Time: 19.1s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \left(\left(\left(1 + x\right) \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x \cdot x - 1}\right) \cdot \frac{1}{x - 1}\right)}{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right) + \frac{1}{x - 1} \cdot \frac{1}{x - 1}}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \left(\left(\left(1 + x\right) \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x \cdot x - 1}\right) \cdot \frac{1}{x - 1}\right)}{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right) + \frac{1}{x - 1} \cdot \frac{1}{x - 1}}
double f(double x) {
        double r3978681 = 1.0;
        double r3978682 = x;
        double r3978683 = r3978682 - r3978681;
        double r3978684 = r3978681 / r3978683;
        double r3978685 = r3978682 + r3978681;
        double r3978686 = r3978682 / r3978685;
        double r3978687 = r3978684 + r3978686;
        return r3978687;
}
↓
double f(double x) {
        double r3978688 = x;
        double r3978689 = 1.0;
        double r3978690 = r3978689 + r3978688;
        double r3978691 = r3978688 / r3978690;
        double r3978692 = r3978691 * r3978691;
        double r3978693 = r3978688 - r3978689;
        double r3978694 = r3978689 / r3978693;
        double r3978695 = r3978690 * r3978694;
        double r3978696 = r3978688 * r3978688;
        double r3978697 = r3978696 - r3978689;
        double r3978698 = r3978689 / r3978697;
        double r3978699 = r3978695 * r3978698;
        double r3978700 = r3978699 * r3978694;
        double r3978701 = fma(r3978692, r3978691, r3978700);
        double r3978702 = r3978694 * r3978691;
        double r3978703 = r3978692 - r3978702;
        double r3978704 = r3978694 * r3978694;
        double r3978705 = r3978703 + r3978704;
        double r3978706 = r3978701 / r3978705;
        return r3978706;
}

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 flip3-+0.0

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

  4. Simplified0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-/r/0.0

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

  8. Applied associate-*l*0.0

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

  9. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \left(\left(\left(1 + x\right) \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x \cdot x - 1}\right) \cdot \frac{1}{x - 1}\right)}{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right) + \frac{1}{x - 1} \cdot \frac{1}{x - 1}}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))