Average Error: 14.9 → 0.4
Time: 2.2s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.7387131770065571 \cdot 10^{-301} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq 0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.7387131770065571 \cdot 10^{-301} \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}\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= (/ x (+ (* x x) 1.0)) -1.7387131770065571e-301)
         (not (<= (/ x (+ (* x x) 1.0)) 0.0)))
   (/ (/ x (sqrt (+ (* x x) 1.0))) (sqrt (+ (* x x) 1.0)))
   (/ 1.0 x)))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (((x / ((x * x) + 1.0)) <= -1.7387131770065571e-301) || !((x / ((x * x) + 1.0)) <= 0.0)) {
		tmp = (x / sqrt((x * x) + 1.0)) / sqrt((x * x) + 1.0);
	} else {
		tmp = 1.0 / x;
	}
	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
Herbie0.4
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (+.f64 (*.f64 x x) 1)) < -1.7387131770065571e-301 or 0.0 < (/.f64 x (+.f64 (*.f64 x x) 1))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_11230.1

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

    4. Applied associate-/r*_binary64_10450.0

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

    if -1.7387131770065571e-301 < (/.f64 x (+.f64 (*.f64 x x) 1)) < 0.0

    1. Initial program 58.2

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

    2. Taylor expanded around inf 1.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.7387131770065571 \cdot 10^{-301} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq 0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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