Average Error: 15.2 → 0.0
Time: 2.6s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{x} - {x}^{-3}\\ \mathbf{if}\;x \leq -14973.991651652683:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 26873.75612472532:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
t_0 := \frac{1}{x} - {x}^{-3}\\
\mathbf{if}\;x \leq -14973.991651652683:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 26873.75612472532:\\
\;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 x) (pow x -3.0))))
   (if (<= x -14973.991651652683)
     t_0
     (if (<= x 26873.75612472532) (* x (/ 1.0 (fma x x 1.0))) t_0))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double t_0 = (1.0 / x) - pow(x, -3.0);
	double tmp;
	if (x <= -14973.991651652683) {
		tmp = t_0;
	} else if (x <= 26873.75612472532) {
		tmp = x * (1.0 / fma(x, x, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -14973.991651652683 or 26873.7561247253216 < x

    1. Initial program 30.7

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

    2. Simplified30.7

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

    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 -14973.991651652683 < x < 26873.7561247253216

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14973.991651652683:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{elif}\;x \leq 26873.75612472532:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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