Average Error: 29.0 → 0.2
Time: 3.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[-\frac{\frac{\mathsf{fma}\left(3, x, 1\right)}{x + 1}}{x - 1}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
-\frac{\frac{\mathsf{fma}\left(3, x, 1\right)}{x + 1}}{x - 1}
double f(double x) {
        double r140310 = x;
        double r140311 = 1.0;
        double r140312 = r140310 + r140311;
        double r140313 = r140310 / r140312;
        double r140314 = r140310 - r140311;
        double r140315 = r140312 / r140314;
        double r140316 = r140313 - r140315;
        return r140316;
}
↓
double f(double x) {
        double r140317 = 3.0;
        double r140318 = x;
        double r140319 = 1.0;
        double r140320 = fma(r140317, r140318, r140319);
        double r140321 = r140318 + r140319;
        double r140322 = r140320 / r140321;
        double r140323 = r140318 - r140319;
        double r140324 = r140322 / r140323;
        double r140325 = -r140324;
        return r140325;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.0

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

  3. Applied frac-sub30.1

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

  4. Simplified30.1

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

  5. Taylor expanded around 0 14.8

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

  7. Applied distribute-frac-neg14.8

    \[\leadsto \color{blue}{-\frac{3 \cdot x + 1}{x \cdot x - 1 \cdot 1}}\]

  8. Simplified0.2

    \[\leadsto -\color{blue}{\frac{\frac{\mathsf{fma}\left(3, x, 1\right)}{x + 1}}{x - 1}}\]

  9. Final simplification0.2

    \[\leadsto -\frac{\frac{\mathsf{fma}\left(3, x, 1\right)}{x + 1}}{x - 1}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))