Average Error: 0.0 → 0.0
Time: 5.4s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\mathsf{fma}\left(\frac{1}{x \cdot x - 1}, x + 1, \frac{x}{x + 1}\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\mathsf{fma}\left(\frac{1}{x \cdot x - 1}, x + 1, \frac{x}{x + 1}\right)
double f(double x) {
        double r3629638 = 1.0;
        double r3629639 = x;
        double r3629640 = r3629639 - r3629638;
        double r3629641 = r3629638 / r3629640;
        double r3629642 = r3629639 + r3629638;
        double r3629643 = r3629639 / r3629642;
        double r3629644 = r3629641 + r3629643;
        return r3629644;
}
↓
double f(double x) {
        double r3629645 = 1.0;
        double r3629646 = x;
        double r3629647 = r3629646 * r3629646;
        double r3629648 = r3629647 - r3629645;
        double r3629649 = r3629645 / r3629648;
        double r3629650 = r3629646 + r3629645;
        double r3629651 = r3629646 / r3629650;
        double r3629652 = fma(r3629649, r3629650, r3629651);
        return r3629652;
}

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 flip--0.0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{x}{x + 1}\]

  4. Applied associate-/r/0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))