Average Error: 15.1 → 0.5
Time: 1.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.8310533803042086 \cdot 10^{-298} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq -0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.8310533803042086 \cdot 10^{-298} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq -0\right):\\
\;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= (/ x (+ (* x x) 1.0)) -1.8310533803042086e-298)
         (not (<= (/ x (+ (* x x) 1.0)) -0.0)))
   (/ (/ x (sqrt (+ (* x x) 1.0))) (sqrt (+ (* x x) 1.0)))
   (+ (/ 1.0 (pow x 5.0)) (- (/ 1.0 x) (/ 1.0 (pow x 3.0))))))
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	double tmp;
	if ((((x / ((double) (((double) (x * x)) + 1.0))) <= -1.8310533803042086e-298) || !((x / ((double) (((double) (x * x)) + 1.0))) <= -0.0))) {
		tmp = ((x / ((double) sqrt(((double) (((double) (x * x)) + 1.0))))) / ((double) sqrt(((double) (((double) (x * x)) + 1.0)))));
	} else {
		tmp = ((double) ((1.0 / ((double) pow(x, 5.0))) + ((double) ((1.0 / x) - (1.0 / ((double) 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

Original15.1
Target0.1
Herbie0.5
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (+ (* x x) 1.0)) < -1.83105338030420862e-298 or -0.0 < (/ x (+ (* x x) 1.0))

    1. Initial program Error: 0.1 bits

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

    3. Applied add-sqr-sqrtError: 0.1 bits

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

    4. Applied associate-/r*Error: 0.0 bits

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

    if -1.83105338030420862e-298 < (/ x (+ (* x x) 1.0)) < -0.0

    1. Initial program Error: 57.8 bits

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

    2. Taylor expanded around inf Error: 1.9 bits

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

    3. SimplifiedError: 1.9 bits

      \[\leadsto \color{blue}{\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.8310533803042086 \cdot 10^{-298} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq -0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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