Average Error: 15.3 → 0.0
Time: 1.3s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2963367744.0775437:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 258616174.30785006:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2963367744.0775437:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{elif}\;x \leq 258616174.30785006:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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


\end{array}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2963367744.0775437)
   (/ 1.0 x)
   (if (<= x 258616174.30785006) (/ x (fma x x 1.0)) (/ 1.0 x))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (x <= -2963367744.0775437) {
		tmp = 1.0 / x;
	} else if (x <= 258616174.30785006) {
		tmp = x / fma(x, x, 1.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -2963367744.07754374 or 258616174.30785006 < x

    1. Initial program 30.6

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

    2. Simplified30.6

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

    3. Taylor expanded in x around inf 0

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

    if -2963367744.07754374 < x < 258616174.30785006

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2963367744.0775437:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 258616174.30785006:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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