Average Error: 28.7 → 0.1
Time: 17.6s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13946.27375560417021915782243013381958008 \lor \neg \left(x \le 10454.06679094586434075608849525451660156\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13946.27375560417021915782243013381958008 \lor \neg \left(x \le 10454.06679094586434075608849525451660156\right):\\
\;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\

\end{array}
double f(double x) {
        double r66070 = x;
        double r66071 = 1.0;
        double r66072 = r66070 + r66071;
        double r66073 = r66070 / r66072;
        double r66074 = r66070 - r66071;
        double r66075 = r66072 / r66074;
        double r66076 = r66073 - r66075;
        return r66076;
}
↓
double f(double x) {
        double r66077 = x;
        double r66078 = -13946.27375560417;
        bool r66079 = r66077 <= r66078;
        double r66080 = 10454.066790945864;
        bool r66081 = r66077 <= r66080;
        double r66082 = !r66081;
        bool r66083 = r66079 || r66082;
        double r66084 = 3.0;
        double r66085 = 3.0;
        double r66086 = pow(r66077, r66085);
        double r66087 = r66084 / r66086;
        double r66088 = -r66087;
        double r66089 = r66084 / r66077;
        double r66090 = 1.0;
        double r66091 = r66090 / r66077;
        double r66092 = r66091 / r66077;
        double r66093 = r66089 + r66092;
        double r66094 = r66088 - r66093;
        double r66095 = 1.0;
        double r66096 = r66077 + r66090;
        double r66097 = r66095 / r66096;
        double r66098 = r66077 - r66090;
        double r66099 = r66096 / r66098;
        double r66100 = -r66099;
        double r66101 = fma(r66077, r66097, r66100);
        double r66102 = r66083 ? r66094 : r66101;
        return r66102;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13946.27375560417 or 10454.066790945864 < x

    1. Initial program 59.4

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Simplified59.4

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

    3. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(1 \cdot \frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    4. Simplified0.0

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

    if -13946.27375560417 < x < 10454.066790945864

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Simplified0.1

      \[\leadsto \color{blue}{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\]
    3. Using strategy rm

    4. Applied div-inv0.1

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

    5. Applied fma-neg0.1

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

    6. Simplified0.1

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{1 + x}, \color{blue}{\frac{-\left(1 + x\right)}{x - 1}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13946.27375560417021915782243013381958008 \lor \neg \left(x \le 10454.06679094586434075608849525451660156\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))