Average Error: 14.3 → 0.0
Time: 15.1s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6935920363808791.0:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{1}{x \cdot x}, {x}^{-2}\right)\\ \mathbf{elif}\;x \le 229105.08111828775:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{1}{x \cdot x}, {x}^{-2}\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6935920363808791.0:\\
\;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{1}{x \cdot x}, {x}^{-2}\right)\\

\mathbf{elif}\;x \le 229105.08111828775:\\
\;\;\;\;\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}\\

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

\end{array}
double f(double x) {
        double r1589761 = 1.0;
        double r1589762 = x;
        double r1589763 = r1589762 + r1589761;
        double r1589764 = r1589761 / r1589763;
        double r1589765 = r1589761 / r1589762;
        double r1589766 = r1589764 - r1589765;
        return r1589766;
}
↓
double f(double x) {
        double r1589767 = x;
        double r1589768 = -6935920363808791.0;
        bool r1589769 = r1589767 <= r1589768;
        double r1589770 = 1.0;
        double r1589771 = r1589767 * r1589767;
        double r1589772 = r1589771 * r1589767;
        double r1589773 = r1589770 / r1589772;
        double r1589774 = r1589770 / r1589771;
        double r1589775 = -2.0;
        double r1589776 = pow(r1589767, r1589775);
        double r1589777 = fma(r1589774, r1589774, r1589776);
        double r1589778 = r1589773 - r1589777;
        double r1589779 = 229105.08111828775;
        bool r1589780 = r1589767 <= r1589779;
        double r1589781 = r1589770 + r1589767;
        double r1589782 = r1589767 - r1589781;
        double r1589783 = r1589767 * r1589781;
        double r1589784 = r1589782 / r1589783;
        double r1589785 = r1589780 ? r1589784 : r1589778;
        double r1589786 = r1589769 ? r1589778 : r1589785;
        return r1589786;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -6935920363808791.0 or 229105.08111828775 < x

    1. Initial program 28.7

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

    3. Applied frac-sub28.0

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

    4. Simplified28.0

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

    5. Simplified28.0

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

    6. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)}\]

    7. Simplified0.8

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

    9. Applied pow10.8

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

    10. Applied pow10.8

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

    11. Applied pow-prod-up0.8

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

    12. Applied pow-flip0.0

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

    13. Simplified0.0

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

    if -6935920363808791.0 < x < 229105.08111828775

    1. Initial program 0.7

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

    3. Applied frac-sub0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6935920363808791.0:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{1}{x \cdot x}, {x}^{-2}\right)\\ \mathbf{elif}\;x \le 229105.08111828775:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot x} - \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{1}{x \cdot x}, {x}^{-2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))