Average Error: 14.6 → 0.0
Time: 3.2s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -20238.151029596927 \lor \neg \left(x \leq 44972.89088513332\right):\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + {x}^{6}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -20238.151029596927 \lor \neg \left(x \leq 44972.89088513332\right):\\
\;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + {x}^{6}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -20238.151029596927) (not (<= x 44972.89088513332)))
   (- (/ 1.0 x) (/ 1.0 (pow x 3.0)))
   (* (/ x (+ 1.0 (pow x 6.0))) (+ (* (* x x) (* x x)) (- 1.0 (* x x))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -20238.151029596927) || !(x <= 44972.89088513332)) {
		tmp = (1.0 / x) - (1.0 / pow(x, 3.0));
	} else {
		tmp = (x / (1.0 + pow(x, 6.0))) * (((x * x) * (x * x)) + (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

Original14.6
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -20238.151029596927 or 44972.8908851333181 < x

    1. Initial program 29.9

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}}\]

    if -20238.151029596927 < x < 44972.8908851333181

    1. Initial program 0.0

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

    3. Applied flip3-+_binary64_11040.0

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

    4. Applied associate-/r/_binary64_10470.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{1 + {x}^{6}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20238.151029596927 \lor \neg \left(x \leq 44972.89088513332\right):\\ \;\;\;\;\frac{1}{x} - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + {x}^{6}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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