Average Error: 14.9 → 0.0
Time: 766.0ms
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.606333490009771 \cdot 10^{+18}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{elif}\;x \leq 5710.861911212298:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.606333490009771 \cdot 10^{+18}:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

\mathbf{elif}\;x \leq 5710.861911212298:\\
\;\;\;\;\frac{x}{x \cdot x + 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -4.606333490009771e+18)
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (/ 1.0 (pow x 3.0)))
   (if (<= x 5710.861911212298)
     (/ x (+ (* x x) 1.0))
     (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (/ 1.0 (pow x 3.0))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (x <= -4.606333490009771e+18) {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - (1.0 / pow(x, 3.0));
	} else if (x <= 5710.861911212298) {
		tmp = x / ((x * x) + 1.0);
	} else {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - (1.0 / pow(x, 3.0));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4606333490009771000 or 5710.861911212298 < x

    1. Initial program 31.0

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

    2. Taylor expanded around inf 0.0

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

    if -4606333490009771000 < x < 5710.861911212298

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.606333490009771 \cdot 10^{+18}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{elif}\;x \leq 5710.861911212298:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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