Average Error: 14.8 → 0.0
Time: 1.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -34509.96394582827:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{elif}\;x \leq 7513.27366378881:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -34509.96394582827:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\

\mathbf{elif}\;x \leq 7513.27366378881:\\
\;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -34509.96394582827)
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (pow (/ 1.0 x) 3.0))
   (if (<= x 7513.27366378881)
     (/ (/ x (sqrt (+ 1.0 (* x x)))) (sqrt (+ 1.0 (* x x))))
     (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (pow (/ 1.0 x) 3.0)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (x <= -34509.96394582827) {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - pow((1.0 / x), 3.0);
	} else if (x <= 7513.27366378881) {
		tmp = (x / sqrt(1.0 + (x * x))) / sqrt(1.0 + (x * x));
	} else {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - pow((1.0 / 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

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -34509.9639458282691 or 7513.2736637888102 < x

    1. Initial program 30.1

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -34509.9639458282691 < x < 7513.2736637888102

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary640.0

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

    4. Applied associate-/r*_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34509.96394582827:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{elif}\;x \leq 7513.27366378881:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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