Average Error: 14.6 → 0.0
Time: 2.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -41237071864426.9453 \lor \neg \left(x \le 12824.6947318943476\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}}, \frac{1}{x} - 1 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -41237071864426.9453 \lor \neg \left(x \le 12824.6947318943476\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}}, \frac{1}{x} - 1 \cdot \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x \cdot x + 1}\\

\end{array}
double code(double x) {
	return (x / ((x * x) + 1.0));
}
↓
double code(double x) {
	double temp;
	if (((x <= -41237071864426.945) || !(x <= 12824.694731894348))) {
		temp = fma(1.0, (1.0 / pow(x, 5.0)), ((1.0 / x) - (1.0 * (1.0 / pow(x, 3.0)))));
	} else {
		temp = (x / ((x * x) + 1.0));
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -41237071864426.945 or 12824.694731894348 < x

    1. Initial program 30.3

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

    3. Applied add-sqr-sqrt30.3

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied *-un-lft-identity30.3

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\]

    5. Applied times-frac30.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}}\]

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -41237071864426.945 < x < 12824.694731894348

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -41237071864426.9453 \lor \neg \left(x \le 12824.6947318943476\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}}, \frac{1}{x} - 1 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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