Average Error: 15.1 → 0.0
Time: 1.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9.836872591914602 \cdot 10^{+26} \lor \neg \left(x \leq 58382.894616906495\right):\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9.836872591914602 \cdot 10^{+26} \lor \neg \left(x \leq 58382.894616906495\right):\\
\;\;\;\;\frac{1}{x} - \frac{1}{{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 -9.836872591914602e+26) (not (<= x 58382.894616906495)))
   (- (/ 1.0 x) (/ 1.0 (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 <= -9.836872591914602e+26) || !(x <= 58382.894616906495)) {
		tmp = (1.0 / x) - (1.0 / 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 < -9.8368725919146019e26 or 58382.8946169064948 < x

    1. Initial program 31.9

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary6431.9

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied associate-/r*_binary6431.8

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]

    5. Simplified31.8

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{1 + x \cdot x}}}}{\sqrt{x \cdot x + 1}}\]

    6. Taylor expanded around inf 0.0

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

    if -9.8368725919146019e26 < x < 58382.8946169064948

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.836872591914602 \cdot 10^{+26} \lor \neg \left(x \leq 58382.894616906495\right):\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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