Average Error: 14.9 → 0.0
Time: 2.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -20496344444626.17:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{elif}\;x \leq 456.806429230397:\\ \;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(-1 + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -20496344444626.17:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\

\mathbf{elif}\;x \leq 456.806429230397:\\
\;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(-1 + x \cdot x\right)\\

\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 -20496344444626.17)
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (pow (/ 1.0 x) 3.0))
   (if (<= x 456.806429230397)
     (* (/ x (+ (pow x 4.0) -1.0)) (+ -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 <= -20496344444626.17) {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - pow((1.0 / x), 3.0);
	} else if (x <= 456.806429230397) {
		tmp = (x / (pow(x, 4.0) + -1.0)) * (-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

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

Derivation

  1. Split input into 2 regimes

  2. if x < -20496344444626.172 or 456.806429230396986 < x

    1. Initial program 30.4

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

    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 -20496344444626.172 < x < 456.806429230396986

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20496344444626.17:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{elif}\;x \leq 456.806429230397:\\ \;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(-1 + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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