Average Error: 14.8 → 0.1
Time: 18.8s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -219.2555067559826227352459682151675224304 \lor \neg \left(x \le 225.8483742386996198092674603685736656189\right):\\ \;\;\;\;-\left(\left(\frac{\frac{2}{x}}{x} + \frac{2}{{x}^{4}}\right) + \frac{2}{{x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{1}^{3} + {x}^{3}}, \mathsf{fma}\left(x, x, 1 \cdot \left(1 - x\right)\right), \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right) \cdot \frac{1}{{x}^{3} - {1}^{3}}\right) + \frac{1}{{x}^{3} - {1}^{3}} \cdot \left(\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right) + \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right)\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -219.2555067559826227352459682151675224304 \lor \neg \left(x \le 225.8483742386996198092674603685736656189\right):\\
\;\;\;\;-\left(\left(\frac{\frac{2}{x}}{x} + \frac{2}{{x}^{4}}\right) + \frac{2}{{x}^{6}}\right)\\

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

\end{array}
double f(double x) {
        double r98461 = 1.0;
        double r98462 = x;
        double r98463 = r98462 + r98461;
        double r98464 = r98461 / r98463;
        double r98465 = r98462 - r98461;
        double r98466 = r98461 / r98465;
        double r98467 = r98464 - r98466;
        return r98467;
}
↓
double f(double x) {
        double r98468 = x;
        double r98469 = -219.25550675598262;
        bool r98470 = r98468 <= r98469;
        double r98471 = 225.84837423869962;
        bool r98472 = r98468 <= r98471;
        double r98473 = !r98472;
        bool r98474 = r98470 || r98473;
        double r98475 = 2.0;
        double r98476 = r98475 / r98468;
        double r98477 = r98476 / r98468;
        double r98478 = 4.0;
        double r98479 = pow(r98468, r98478);
        double r98480 = r98475 / r98479;
        double r98481 = r98477 + r98480;
        double r98482 = 6.0;
        double r98483 = pow(r98468, r98482);
        double r98484 = r98475 / r98483;
        double r98485 = r98481 + r98484;
        double r98486 = -r98485;
        double r98487 = 1.0;
        double r98488 = 3.0;
        double r98489 = pow(r98487, r98488);
        double r98490 = pow(r98468, r98488);
        double r98491 = r98489 + r98490;
        double r98492 = r98487 / r98491;
        double r98493 = r98487 - r98468;
        double r98494 = r98487 * r98493;
        double r98495 = fma(r98468, r98468, r98494);
        double r98496 = r98487 + r98468;
        double r98497 = r98487 * r98496;
        double r98498 = fma(r98468, r98468, r98497);
        double r98499 = -r98498;
        double r98500 = r98490 - r98489;
        double r98501 = r98487 / r98500;
        double r98502 = r98499 * r98501;
        double r98503 = fma(r98492, r98495, r98502);
        double r98504 = r98498 + r98499;
        double r98505 = r98501 * r98504;
        double r98506 = r98503 + r98505;
        double r98507 = r98474 ? r98486 : r98506;
        return r98507;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -219.25550675598262 or 225.84837423869962 < x

    1. Initial program 29.8

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

    4. Taylor expanded around 0 0.7

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

    5. Simplified0.1

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

    if -219.25550675598262 < x < 225.84837423869962

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/r/0.0

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

    5. Applied flip3-+0.0

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

    6. Applied associate-/r/0.0

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

    7. Applied prod-diff0.0

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

    8. Simplified0.0

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

    9. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -219.2555067559826227352459682151675224304 \lor \neg \left(x \le 225.8483742386996198092674603685736656189\right):\\ \;\;\;\;-\left(\left(\frac{\frac{2}{x}}{x} + \frac{2}{{x}^{4}}\right) + \frac{2}{{x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{1}^{3} + {x}^{3}}, \mathsf{fma}\left(x, x, 1 \cdot \left(1 - x\right)\right), \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right) \cdot \frac{1}{{x}^{3} - {1}^{3}}\right) + \frac{1}{{x}^{3} - {1}^{3}} \cdot \left(\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right) + \left(-\mathsf{fma}\left(x, x, 1 \cdot \left(1 + x\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))