Average Error: 14.8 → 0.0
Time: 1.8s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -15247.443439880883 \lor \neg \left(x \leq 471.0555901705247\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(-1 + x \cdot x\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -15247.443439880883 \lor \neg \left(x \leq 471.0555901705247\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {x}^{-3}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -15247.443439880883) (not (<= x 471.0555901705247)))
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (pow x -3.0))
   (* (/ x (+ (pow x 4.0) -1.0)) (+ -1.0 (* x x)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -15247.443439880883) || !(x <= 471.0555901705247)) {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - pow(x, -3.0);
	} else {
		tmp = (x / (pow(x, 4.0) + -1.0)) * (-1.0 + (x * x));
	}
	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 < -15247.4434398808826 or 471.055590170524681 < x

    1. Initial program 30.0

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

    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

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

    5. Applied inv-pow_binary64_18680.0

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

    6. Applied pow-pow_binary64_18550.0

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

    7. Simplified0.0

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

    if -15247.4434398808826 < x < 471.055590170524681

    1. Initial program 0.0

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

    3. Applied flip-+_binary64_17570.0

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

    4. Applied associate-/r/_binary64_17290.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15247.443439880883 \lor \neg \left(x \leq 471.0555901705247\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(-1 + x \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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