Average Error: 14.8 → 0.0
Time: 1.1s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -8.11969149086397 \cdot 10^{+37} \lor \neg \left(x \leq 2751642470.767679\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -8.11969149086397 \cdot 10^{+37} \lor \neg \left(x \leq 2751642470.767679\right):\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\


\end{array}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -8.11969149086397e+37) (not (<= x 2751642470.767679)))
   (/ 1.0 x)
   (expm1 (log1p (/ x (fma x x 1.0))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -8.11969149086397e+37) || !(x <= 2751642470.767679)) {
		tmp = 1.0 / x;
	} else {
		tmp = expm1(log1p(x / fma(x, x, 1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -8.11969149086396955e37 or 2751642470.7676792 < x

    1. Initial program 32.0

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

    2. Simplified32.0

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

    3. Taylor expanded in x around inf 0

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

    if -8.11969149086396955e37 < x < 2751642470.7676792

    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. Applied expm1-log1p-u_binary640.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.11969149086397 \cdot 10^{+37} \lor \neg \left(x \leq 2751642470.767679\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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