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 r1997403 = 1.0;
        double r1997404 = x;
        double r1997405 = r1997404 + r1997403;
        double r1997406 = r1997403 / r1997405;
        double r1997407 = r1997403 / r1997404;
        double r1997408 = r1997406 - r1997407;
        return r1997408;
}
↓
double f(double x) {
        double r1997409 = x;
        double r1997410 = -6935920363808791.0;
        bool r1997411 = r1997409 <= r1997410;
        double r1997412 = 1.0;
        double r1997413 = r1997409 * r1997409;
        double r1997414 = r1997413 * r1997409;
        double r1997415 = r1997412 / r1997414;
        double r1997416 = r1997412 / r1997413;
        double r1997417 = -2.0;
        double r1997418 = pow(r1997409, r1997417);
        double r1997419 = fma(r1997416, r1997416, r1997418);
        double r1997420 = r1997415 - r1997419;
        double r1997421 = 229105.08111828775;
        bool r1997422 = r1997409 <= r1997421;
        double r1997423 = r1997412 + r1997409;
        double r1997424 = r1997409 - r1997423;
        double r1997425 = r1997409 * r1997423;
        double r1997426 = r1997424 / r1997425;
        double r1997427 = r1997422 ? r1997426 : r1997420;
        double r1997428 = r1997411 ? r1997420 : r1997427;
        return r1997428;
}

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

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

    9. Applied pow10.8

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

    10. Applied pow10.8

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

    11. Applied pow-prod-up0.8

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

    12. Applied pow-flip0.0

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

    13. Simplified0.0

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

    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)))