Average Error: 15.1 → 0.0
Time: 3.8s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4179606.248108378 \lor \neg \left(x \leq 12478.320956755231\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4179606.248108378 \lor \neg \left(x \leq 12478.320956755231\right):\\
\;\;\;\;\frac{1}{x} - {x}^{-3}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -4179606.248108378) (not (<= x 12478.320956755231)))
   (- (/ 1.0 x) (pow x -3.0))
   (/ x (+ 1.0 (* x x)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -4179606.248108378) || !(x <= 12478.320956755231)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -4179606.24810837815 or 12478.3209567552312 < x

    1. Initial program 30.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}}\]
    3. Using strategy rm

    4. Applied pow-flip_binary64_11750.0

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

    if -4179606.24810837815 < x < 12478.3209567552312

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4179606.248108378 \lor \neg \left(x \leq 12478.320956755231\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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