Average Error: 29.7 → 0.1
Time: 5.7m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6263.833804380613:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7862.0498067640765:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6263.833804380613:\\
\;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 7862.0498067640765:\\
\;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\

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

\end{array}
double f(double x) {
        double r2536917 = x;
        double r2536918 = 1.0;
        double r2536919 = r2536917 + r2536918;
        double r2536920 = r2536917 / r2536919;
        double r2536921 = r2536917 - r2536918;
        double r2536922 = r2536919 / r2536921;
        double r2536923 = r2536920 - r2536922;
        return r2536923;
}
↓
double f(double x) {
        double r2536924 = x;
        double r2536925 = -6263.833804380613;
        bool r2536926 = r2536924 <= r2536925;
        double r2536927 = -3.0;
        double r2536928 = r2536927 / r2536924;
        double r2536929 = -1.0;
        double r2536930 = r2536924 * r2536924;
        double r2536931 = r2536929 / r2536930;
        double r2536932 = r2536928 + r2536931;
        double r2536933 = r2536930 * r2536924;
        double r2536934 = r2536927 / r2536933;
        double r2536935 = r2536932 + r2536934;
        double r2536936 = 7862.0498067640765;
        bool r2536937 = r2536924 <= r2536936;
        double r2536938 = 1.0;
        double r2536939 = r2536924 + r2536938;
        double r2536940 = r2536930 + r2536939;
        double r2536941 = -r2536940;
        double r2536942 = r2536938 + r2536924;
        double r2536943 = 3.0;
        double r2536944 = pow(r2536924, r2536943);
        double r2536945 = r2536944 - r2536938;
        double r2536946 = r2536942 / r2536945;
        double r2536947 = r2536946 * r2536940;
        double r2536948 = fma(r2536941, r2536946, r2536947);
        double r2536949 = r2536924 / r2536942;
        double r2536950 = cbrt(r2536949);
        double r2536951 = r2536950 * r2536950;
        double r2536952 = r2536946 * r2536941;
        double r2536953 = fma(r2536951, r2536950, r2536952);
        double r2536954 = r2536948 + r2536953;
        double r2536955 = r2536937 ? r2536954 : r2536935;
        double r2536956 = r2536926 ? r2536935 : r2536955;
        return r2536956;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -6263.833804380613 or 7862.0498067640765 < x

    1. Initial program 59.3

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

    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 -6263.833804380613 < x < 7862.0498067640765

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-cube-cbrt0.1

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

    6. Applied prod-diff0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6263.833804380613:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 7862.0498067640765:\\ \;\;\;\;\mathsf{fma}\left(-\left(x \cdot x + \left(x + 1\right)\right), \frac{1 + x}{{x}^{3} - 1}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(x + 1\right)\right)\right) + \mathsf{fma}\left(\sqrt[3]{\frac{x}{1 + x}} \cdot \sqrt[3]{\frac{x}{1 + x}}, \sqrt[3]{\frac{x}{1 + x}}, \frac{1 + x}{{x}^{3} - 1} \cdot \left(-\left(x \cdot x + \left(x + 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{-3}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))