Average Error: 0.0 → 0.0
Time: 7.3s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\log \left(e^{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)}\right) + \frac{x}{x + 1}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\log \left(e^{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)}\right) + \frac{x}{x + 1}
double f(double x) {
        double r2058661 = 1.0;
        double r2058662 = x;
        double r2058663 = r2058662 - r2058661;
        double r2058664 = r2058661 / r2058663;
        double r2058665 = r2058662 + r2058661;
        double r2058666 = r2058662 / r2058665;
        double r2058667 = r2058664 + r2058666;
        return r2058667;
}
↓
double f(double x) {
        double r2058668 = 1.0;
        double r2058669 = x;
        double r2058670 = -1.0;
        double r2058671 = fma(r2058669, r2058669, r2058670);
        double r2058672 = r2058668 / r2058671;
        double r2058673 = r2058669 + r2058668;
        double r2058674 = r2058672 * r2058673;
        double r2058675 = exp(r2058674);
        double r2058676 = log(r2058675);
        double r2058677 = r2058669 / r2058673;
        double r2058678 = r2058676 + r2058677;
        return r2058678;
}

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. Simplified0.0

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

  7. Applied add-log-exp0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))