Average Error: 29.7 → 0.1
Time: 2.7m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7634.784366325685:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7132.613910539861:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(1 + x\right)\right), \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}, \left(x \cdot x + \left(1 + x\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right) + \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \frac{\sqrt[3]{x}}{1 + x}, \left(-\left(x \cdot x + \left(1 + x\right)\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7634.784366325685:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 7132.613910539861:\\
\;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(1 + x\right)\right), \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}, \left(x \cdot x + \left(1 + x\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right) + \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \frac{\sqrt[3]{x}}{1 + x}, \left(-\left(x \cdot x + \left(1 + x\right)\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\

\end{array}
double f(double x) {
        double r4165081 = x;
        double r4165082 = 1.0;
        double r4165083 = r4165081 + r4165082;
        double r4165084 = r4165081 / r4165083;
        double r4165085 = r4165081 - r4165082;
        double r4165086 = r4165083 / r4165085;
        double r4165087 = r4165084 - r4165086;
        return r4165087;
}
↓
double f(double x) {
        double r4165088 = x;
        double r4165089 = -7634.784366325685;
        bool r4165090 = r4165088 <= r4165089;
        double r4165091 = -1.0;
        double r4165092 = r4165088 * r4165088;
        double r4165093 = r4165091 / r4165092;
        double r4165094 = -3.0;
        double r4165095 = r4165094 / r4165088;
        double r4165096 = r4165093 + r4165095;
        double r4165097 = r4165092 * r4165088;
        double r4165098 = r4165094 / r4165097;
        double r4165099 = r4165096 + r4165098;
        double r4165100 = 7132.613910539861;
        bool r4165101 = r4165088 <= r4165100;
        double r4165102 = 1.0;
        double r4165103 = r4165102 + r4165088;
        double r4165104 = r4165092 + r4165103;
        double r4165105 = -r4165104;
        double r4165106 = r4165092 - r4165102;
        double r4165107 = 3.0;
        double r4165108 = pow(r4165088, r4165107);
        double r4165109 = r4165108 - r4165102;
        double r4165110 = r4165088 - r4165102;
        double r4165111 = r4165109 * r4165110;
        double r4165112 = r4165106 / r4165111;
        double r4165113 = r4165104 * r4165112;
        double r4165114 = fma(r4165105, r4165112, r4165113);
        double r4165115 = cbrt(r4165088);
        double r4165116 = r4165115 * r4165115;
        double r4165117 = r4165115 / r4165103;
        double r4165118 = r4165105 * r4165112;
        double r4165119 = fma(r4165116, r4165117, r4165118);
        double r4165120 = r4165114 + r4165119;
        double r4165121 = r4165101 ? r4165120 : r4165099;
        double r4165122 = r4165090 ? r4165099 : r4165121;
        return r4165122;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -7634.784366325685 or 7132.613910539861 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -7634.784366325685 < x < 7132.613910539861

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/l/0.1

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

    6. Applied flip3--0.1

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

    7. Applied associate-*l/0.1

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

    8. Applied associate-/r/0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied add-cube-cbrt0.1

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

    11. Applied times-frac0.1

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

    12. Applied prod-diff0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7634.784366325685:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7132.613910539861:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(1 + x\right)\right), \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}, \left(x \cdot x + \left(1 + x\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right) + \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \frac{\sqrt[3]{x}}{1 + x}, \left(-\left(x \cdot x + \left(1 + x\right)\right)\right) \cdot \frac{x \cdot x - 1}{\left({x}^{3} - 1\right) \cdot \left(x - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

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