Average Error: 14.8 → 0.0
Time: 13.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -34915225796078.01171875 \lor \neg \left(x \le 29660.81474128061745432205498218536376953\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{6} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -34915225796078.01171875 \lor \neg \left(x \le 29660.81474128061745432205498218536376953\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{{x}^{6} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\\

\end{array}
double f(double x) {
        double r43470 = x;
        double r43471 = r43470 * r43470;
        double r43472 = 1.0;
        double r43473 = r43471 + r43472;
        double r43474 = r43470 / r43473;
        return r43474;
}
↓
double f(double x) {
        double r43475 = x;
        double r43476 = -34915225796078.01;
        bool r43477 = r43475 <= r43476;
        double r43478 = 29660.814741280617;
        bool r43479 = r43475 <= r43478;
        double r43480 = !r43479;
        bool r43481 = r43477 || r43480;
        double r43482 = 1.0;
        double r43483 = 5.0;
        double r43484 = pow(r43475, r43483);
        double r43485 = r43482 / r43484;
        double r43486 = 1.0;
        double r43487 = r43486 / r43475;
        double r43488 = 3.0;
        double r43489 = pow(r43475, r43488);
        double r43490 = r43482 / r43489;
        double r43491 = r43487 - r43490;
        double r43492 = r43485 + r43491;
        double r43493 = 6.0;
        double r43494 = pow(r43475, r43493);
        double r43495 = pow(r43482, r43488);
        double r43496 = r43494 + r43495;
        double r43497 = r43475 / r43496;
        double r43498 = r43475 * r43475;
        double r43499 = r43498 * r43498;
        double r43500 = r43482 * r43482;
        double r43501 = r43498 * r43482;
        double r43502 = r43500 - r43501;
        double r43503 = r43499 + r43502;
        double r43504 = r43497 * r43503;
        double r43505 = r43481 ? r43492 : r43504;
        return r43505;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -34915225796078.01 or 29660.814741280617 < x

    1. Initial program 30.8

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -34915225796078.01 < x < 29660.814741280617

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{{x}^{6} + {1}^{3}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -34915225796078.01171875 \lor \neg \left(x \le 29660.81474128061745432205498218536376953\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{6} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))