Average Error: 15.0 → 0.0
Time: 883.0ms
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -31947494282464:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 38360308727927630:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -31947494282464:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{elif}\;x \leq 38360308727927630:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

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


\end{array}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -31947494282464.0)
   (/ 1.0 x)
   (if (<= x 38360308727927630.0) (/ x (+ 1.0 (* x x))) (/ 1.0 x))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (x <= -31947494282464.0) {
		tmp = 1.0 / x;
	} else if (x <= 38360308727927630.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -31947494282464 or 38360308727927632 < x

    1. Initial program 31.4

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

    2. Simplified31.4

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}} \]

    3. Taylor expanded in x around inf 0

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

    if -31947494282464 < x < 38360308727927632

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31947494282464:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 38360308727927630:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022121 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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