Average Error: 0.0 → 0.0
Time: 25.7s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}\right), \left(x + 1\right), \left(\frac{x}{x + 1}\right)\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\mathsf{fma}\left(\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}\right), \left(x + 1\right), \left(\frac{x}{x + 1}\right)\right)
double f(double x) {
        double r11868256 = 1.0;
        double r11868257 = x;
        double r11868258 = r11868257 - r11868256;
        double r11868259 = r11868256 / r11868258;
        double r11868260 = r11868257 + r11868256;
        double r11868261 = r11868257 / r11868260;
        double r11868262 = r11868259 + r11868261;
        return r11868262;
}
↓
double f(double x) {
        double r11868263 = 1.0;
        double r11868264 = x;
        double r11868265 = -1.0;
        double r11868266 = fma(r11868264, r11868264, r11868265);
        double r11868267 = r11868263 / r11868266;
        double r11868268 = r11868264 + r11868263;
        double r11868269 = r11868264 / r11868268;
        double r11868270 = fma(r11868267, r11868268, r11868269);
        return r11868270;
}

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

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))