Average Error: 15.3 → 0.0
Time: 1.1s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.3254608192918197 \cdot 10^{+59} \lor \neg \left(x \leq 105927142.50629535\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.3254608192918197 \cdot 10^{+59} \lor \neg \left(x \leq 105927142.50629535\right):\\
\;\;\;\;\frac{1}{x}\\

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


\end{array}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -3.3254608192918197e+59) (not (<= x 105927142.50629535)))
   (/ 1.0 x)
   (/ x (fma x x 1.0))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -3.3254608192918197e+59) || !(x <= 105927142.50629535)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / fma(x, x, 1.0);
	}
	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 < -3.3254608192918197e59 or 105927142.50629535 < x

    1. Initial program 34.4

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

    2. Simplified34.4

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

    3. Taylor expanded in x around inf 0

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

    if -3.3254608192918197e59 < x < 105927142.50629535

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3254608192918197 \cdot 10^{+59} \lor \neg \left(x \leq 105927142.50629535\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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