Average Error: 15.0 → 0.0
Time: 1.3min
Precision: binary64
Cost: 1284
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -495675560.2626342 \lor \neg \left(x \leq 439.8864172703267\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -495675560.2626342 \lor \neg \left(x \leq 439.8864172703267\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -495675560.2626342) (not (<= x 439.8864172703267)))
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (/ 1.0 (pow x 3.0)))
   (* x (/ 1.0 (+ 1.0 (* x x))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -495675560.2626342) || !(x <= 439.8864172703267)) {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - (1.0 / pow(x, 3.0));
	} else {
		tmp = x * (1.0 / (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

Original15.0
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]
Alternative 1
Accuracy0.0
Cost2052
\[\begin{array}{l} \mathbf{if}\;x \leq -3.059179128579821 \cdot 10^{+22} \lor \neg \left(x \leq 3711.205976017636\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\left|\frac{\sqrt{1 + {x}^{6}}}{\sqrt{{x}^{4} + \left(1 - x \cdot x\right)}}\right|}\\ \end{array}\]
Alternative 2
Accuracy15.0
Cost960
\[\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\]
Alternative 3
Accuracy15.1
Cost576
\[\frac{1}{\frac{1 + x \cdot x}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -495675560.262634218 or 439.886417270326717 < x

    1. Initial program 30.5

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

    3. Applied add-sqr-sqrt_binary64_44130.5

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

    4. Applied associate-/r*_binary64_36330.4

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

    5. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}}\]

    if -495675560.262634218 < x < 439.886417270326717

    1. Initial program 0.0

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

    3. Applied div-inv_binary64_4160.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -495675560.2626342 \lor \neg \left(x \leq 439.8864172703267\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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