Average Error: 15.4 → 0.0
Time: 2.2s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -111505787.31416105 \lor \neg \left(x \leq 82504136.52564584\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -111505787.31416105 \lor \neg \left(x \leq 82504136.52564584\right):\\
\;\;\;\;\frac{1}{x}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -111505787.31416105) (not (<= x 82504136.52564584)))
   (/ 1.0 x)
   (* x (/ 1.0 (+ 1.0 (* x x))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -111505787.31416105) || !(x <= 82504136.52564584)) {
		tmp = 1.0 / x;
	} else {
		tmp = x * (1.0 / (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

Original15.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -111505787.314161047 or 82504136.5256458372 < x

    1. Initial program 31.2

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

    2. Taylor expanded around inf 0.0

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

    if -111505787.314161047 < x < 82504136.5256458372

    1. Initial program 0.0

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

    3. Applied div-inv_binary64_10980.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -111505787.31416105 \lor \neg \left(x \leq 82504136.52564584\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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