Average Error: 14.9 → 0.0
Time: 1.2min
Precision: binary64
Cost: 776
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5.176580413214303 \cdot 10^{+23} \lor \neg \left(x \leq 56166425.491830505\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -5.176580413214303 \cdot 10^{+23} \lor \neg \left(x \leq 56166425.491830505\right):\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -5.176580413214303e+23) (not (<= x 56166425.491830505)))
   (/ 1.0 x)
   (/ x (+ 1.0 (* x x)))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -5.176580413214303e+23) || !(x <= 56166425.491830505)) {
		tmp = 1.0 / x;
	} else {
		tmp = 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

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

Alternatives

Alternative 1
Error0.1
Cost448
\[\frac{-1}{\frac{-1}{x} - x}\]
Alternative 2
Error0.6
Cost520
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0278398150984853 \lor \neg \left(x \leq 1.00048819294968\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
Alternative 3
Error31.1
Cost64
\[x\]

Error

Time

Derivation

  1. Split input into 2 regimes

  2. if x < -5.1765804132143032e23 or 56166425.4918305054 < x

    1. Initial program 31.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{x}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{x}}\]

    if -5.1765804132143032e23 < x < 56166425.4918305054

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity_binary64_14420.0

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

    4. Applied *-un-lft-identity_binary64_14420.0

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

    5. Applied times-frac_binary64_14480.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.176580413214303 \cdot 10^{+23} \lor \neg \left(x \leq 56166425.491830505\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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