Average Error: 15.3 → 0.0
Time: 1.1s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.3546673488908415 \cdot 10^{+34} \lor \neg \left(x \leq 90888509.91221009\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.3546673488908415 \cdot 10^{+34} \lor \neg \left(x \leq 90888509.91221009\right):\\
\;\;\;\;\frac{1}{x}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -3.3546673488908415e+34) (not (<= x 90888509.91221009)))
   (/ 1.0 x)
   (/ x (+ 1.0 (* x x)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -3.3546673488908415e+34) || !(x <= 90888509.91221009)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.3546673488908415e34 or 90888509.9122100919 < x

    1. Initial program 32.3

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

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{\frac{1}{x}}\]

    if -3.3546673488908415e34 < x < 90888509.9122100919

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3546673488908415 \cdot 10^{+34} \lor \neg \left(x \leq 90888509.91221009\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))