Average Error: 14.9 → 0.0
Time: 4.5s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -235.018390279293243 \lor \neg \left(x \le 210.68103720548342\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - \mathsf{fma}\left(2, {x}^{\left(-2\right)}, 2 \cdot \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{x}^{3} + {1}^{3}}, x \cdot x + \left(1 \cdot 1 - x \cdot 1\right), -\frac{1}{x - 1}\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -235.018390279293243 \lor \neg \left(x \le 210.68103720548342\right):\\
\;\;\;\;\frac{-2}{{x}^{6}} - \mathsf{fma}\left(2, {x}^{\left(-2\right)}, 2 \cdot \frac{1}{{x}^{4}}\right)\\

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

\end{array}
double f(double x) {
        double r130700 = 1.0;
        double r130701 = x;
        double r130702 = r130701 + r130700;
        double r130703 = r130700 / r130702;
        double r130704 = r130701 - r130700;
        double r130705 = r130700 / r130704;
        double r130706 = r130703 - r130705;
        return r130706;
}
↓
double f(double x) {
        double r130707 = x;
        double r130708 = -235.01839027929324;
        bool r130709 = r130707 <= r130708;
        double r130710 = 210.68103720548342;
        bool r130711 = r130707 <= r130710;
        double r130712 = !r130711;
        bool r130713 = r130709 || r130712;
        double r130714 = 2.0;
        double r130715 = -r130714;
        double r130716 = 6.0;
        double r130717 = pow(r130707, r130716);
        double r130718 = r130715 / r130717;
        double r130719 = 2.0;
        double r130720 = -r130719;
        double r130721 = pow(r130707, r130720);
        double r130722 = 1.0;
        double r130723 = 4.0;
        double r130724 = pow(r130707, r130723);
        double r130725 = r130722 / r130724;
        double r130726 = r130714 * r130725;
        double r130727 = fma(r130714, r130721, r130726);
        double r130728 = r130718 - r130727;
        double r130729 = 1.0;
        double r130730 = 3.0;
        double r130731 = pow(r130707, r130730);
        double r130732 = pow(r130729, r130730);
        double r130733 = r130731 + r130732;
        double r130734 = r130729 / r130733;
        double r130735 = r130707 * r130707;
        double r130736 = r130729 * r130729;
        double r130737 = r130707 * r130729;
        double r130738 = r130736 - r130737;
        double r130739 = r130735 + r130738;
        double r130740 = r130707 - r130729;
        double r130741 = r130729 / r130740;
        double r130742 = -r130741;
        double r130743 = fma(r130734, r130739, r130742);
        double r130744 = r130713 ? r130728 : r130743;
        return r130744;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -235.01839027929324 or 210.68103720548342 < x

    1. Initial program 29.5

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

    2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

    3. Simplified0.8

      \[\leadsto \color{blue}{\frac{-2}{{x}^{6}} - \mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 \cdot \frac{1}{{x}^{4}}\right)}\]
    4. Using strategy rm

    5. Applied pow-flip0.0

      \[\leadsto \frac{-2}{{x}^{6}} - \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 \cdot \frac{1}{{x}^{4}}\right)\]

    if -235.01839027929324 < x < 210.68103720548342

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Applied fma-neg0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -235.018390279293243 \lor \neg \left(x \le 210.68103720548342\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - \mathsf{fma}\left(2, {x}^{\left(-2\right)}, 2 \cdot \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{x}^{3} + {1}^{3}}, x \cdot x + \left(1 \cdot 1 - x \cdot 1\right), -\frac{1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))