Average Error: 14.9 → 0.0
Time: 1.5s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -739594429.2319791 \lor \neg \left(x \leq 860.8941305977887\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -739594429.2319791 \lor \neg \left(x \leq 860.8941305977887\right):\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -739594429.2319791) (not (<= x 860.8941305977887)))
   (- (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (/ 1.0 (pow x 3.0)))
   (* (/ x (- (pow x 4.0) (* 1.0 1.0))) (- (* x x) 1.0))))
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -739594429.2319791) || !(x <= 860.8941305977887))) {
		tmp = ((double) (((double) ((1.0 / ((double) pow(x, 5.0))) + (1.0 / x))) - (1.0 / ((double) pow(x, 3.0)))));
	} else {
		tmp = ((double) ((x / ((double) (((double) pow(x, 4.0)) - ((double) (1.0 * 1.0))))) * ((double) (((double) (x * x)) - 1.0))));
	}
	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 < -739594429.23197913 or 860.89413059778872 < x

    1. Initial program 30.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -739594429.23197913 < x < 860.89413059778872

    1. Initial program 0.0

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

    3. Applied flip-+_binary640.0

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

    4. Applied associate-/r/_binary640.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{{x}^{4} - 1 \cdot 1}} \cdot \left(x \cdot x - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -739594429.2319791 \lor \neg \left(x \leq 860.8941305977887\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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