Average Error: 15.0 → 0.0
Time: 5.3s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -11756.998493388033 \lor \neg \left(x \leq 88225.07844131009\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -11756.998493388033 \lor \neg \left(x \leq 88225.07844131009\right):\\
\;\;\;\;\frac{1}{x} - {x}^{-3}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{\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 -11756.998493388033) (not (<= x 88225.07844131009)))
   (- (/ 1.0 x) (pow x -3.0))
   (* x (/ 1.0 (fma x x 1.0)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -11756.998493388033) || !(x <= 88225.07844131009)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x * (1.0 / fma(x, x, 1.0));
	}
	return tmp;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -11756.998493388033 or 88225.078441310092 < x

    1. Initial program 29.7

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

    2. Simplified29.7

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

    3. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]

    4. Applied pow-flip_binary640.0

      \[\leadsto \frac{1}{x} - \color{blue}{{x}^{\left(-3\right)}} \]

    5. Simplified0.0

      \[\leadsto \frac{1}{x} - {x}^{\color{blue}{-3}} \]

    if -11756.998493388033 < x < 88225.078441310092

    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 div-inv_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11756.998493388033 \lor \neg \left(x \leq 88225.07844131009\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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