Average Error: 10.0 → 0.1
Time: 6.4s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -0.033504988994731283 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.7156110759 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x \cdot x}}{x}\right)\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -0.033504988994731283 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.7156110759 \cdot 10^{-8}\right):\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x \cdot x}}{x}\right)\right)\\

\end{array}
double f(double x) {
        double r150847 = 1.0;
        double r150848 = x;
        double r150849 = r150848 + r150847;
        double r150850 = r150847 / r150849;
        double r150851 = 2.0;
        double r150852 = r150851 / r150848;
        double r150853 = r150850 - r150852;
        double r150854 = r150848 - r150847;
        double r150855 = r150847 / r150854;
        double r150856 = r150853 + r150855;
        return r150856;
}
↓
double f(double x) {
        double r150857 = 1.0;
        double r150858 = x;
        double r150859 = r150858 + r150857;
        double r150860 = r150857 / r150859;
        double r150861 = 2.0;
        double r150862 = r150861 / r150858;
        double r150863 = r150860 - r150862;
        double r150864 = r150858 - r150857;
        double r150865 = r150857 / r150864;
        double r150866 = r150863 + r150865;
        double r150867 = -0.03350498899473128;
        bool r150868 = r150866 <= r150867;
        double r150869 = 1.715611075902776e-08;
        bool r150870 = r150866 <= r150869;
        double r150871 = !r150870;
        bool r150872 = r150868 || r150871;
        double r150873 = 1.0;
        double r150874 = 7.0;
        double r150875 = pow(r150858, r150874);
        double r150876 = r150873 / r150875;
        double r150877 = 5.0;
        double r150878 = pow(r150858, r150877);
        double r150879 = r150873 / r150878;
        double r150880 = r150858 * r150858;
        double r150881 = r150861 / r150880;
        double r150882 = r150881 / r150858;
        double r150883 = fma(r150861, r150879, r150882);
        double r150884 = fma(r150861, r150876, r150883);
        double r150885 = r150872 ? r150866 : r150884;
        return r150885;
}

Error

Bits error versus x

Target

Original10.0
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -0.03350498899473128 or 1.715611075902776e-08 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

    1. Initial program 0.0

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

    if -0.03350498899473128 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 1.715611075902776e-08

    1. Initial program 19.7

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

    5. Applied unpow30.7

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

    6. Applied associate-/r*0.2

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \color{blue}{\frac{\frac{2}{x \cdot x}}{x}}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -0.033504988994731283 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 1.7156110759 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x \cdot x}}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))))