Average Error: 10.2 → 0.1
Time: 7.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -109.218011094796353 \lor \neg \left(x \le 115.29908004264922\right):\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{1}{x \cdot x} \cdot \frac{2}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1}\right)\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -109.218011094796353 \lor \neg \left(x \le 115.29908004264922\right):\\
\;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{1}{x \cdot x} \cdot \frac{2}{x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1}\right)\right)\\

\end{array}
double f(double x) {
        double r168721 = 1.0;
        double r168722 = x;
        double r168723 = r168722 + r168721;
        double r168724 = r168721 / r168723;
        double r168725 = 2.0;
        double r168726 = r168725 / r168722;
        double r168727 = r168724 - r168726;
        double r168728 = r168722 - r168721;
        double r168729 = r168721 / r168728;
        double r168730 = r168727 + r168729;
        return r168730;
}
↓
double f(double x) {
        double r168731 = x;
        double r168732 = -109.21801109479635;
        bool r168733 = r168731 <= r168732;
        double r168734 = 115.29908004264922;
        bool r168735 = r168731 <= r168734;
        double r168736 = !r168735;
        bool r168737 = r168733 || r168736;
        double r168738 = 2.0;
        double r168739 = 1.0;
        double r168740 = 7.0;
        double r168741 = pow(r168731, r168740);
        double r168742 = r168739 / r168741;
        double r168743 = 5.0;
        double r168744 = pow(r168731, r168743);
        double r168745 = r168739 / r168744;
        double r168746 = r168731 * r168731;
        double r168747 = r168739 / r168746;
        double r168748 = r168738 / r168731;
        double r168749 = r168747 * r168748;
        double r168750 = fma(r168738, r168745, r168749);
        double r168751 = fma(r168738, r168742, r168750);
        double r168752 = 1.0;
        double r168753 = r168731 + r168752;
        double r168754 = r168752 / r168753;
        double r168755 = r168754 - r168748;
        double r168756 = r168731 - r168752;
        double r168757 = r168752 / r168756;
        double r168758 = expm1(r168757);
        double r168759 = log1p(r168758);
        double r168760 = r168755 + r168759;
        double r168761 = r168737 ? r168751 : r168760;
        return r168761;
}

Error

Bits error versus x

Target

Original10.2
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -109.21801109479635 or 115.29908004264922 < x

    1. Initial program 19.9

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.2

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}\right)\right)\]

    6. Applied unpow-prod-down1.2

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}\right)\right)\]

    7. Applied *-un-lft-identity1.2

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\color{blue}{1 \cdot 2}}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}\right)\right)\]

    8. Applied times-frac0.8

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \color{blue}{\frac{1}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}} \cdot \frac{2}{{\left(\sqrt[3]{x}\right)}^{3}}}\right)\right)\]

    9. Simplified0.4

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \color{blue}{\frac{1}{x \cdot x}} \cdot \frac{2}{{\left(\sqrt[3]{x}\right)}^{3}}\right)\right)\]

    10. Simplified0.1

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{1}{x \cdot x} \cdot \color{blue}{\frac{2}{x}}\right)\right)\]

    if -109.21801109479635 < x < 115.29908004264922

    1. Initial program 0.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    2. Using strategy rm

    3. Applied log1p-expm1-u0.1

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -109.218011094796353 \lor \neg \left(x \le 115.29908004264922\right):\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{1}{x \cdot x} \cdot \frac{2}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x - 1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))