Average Error: 15.3 → 0.0
Time: 1.7s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -128910779844.2735 \lor \neg \left(x \le 407.785094800499166\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\left(x \cdot x\right)}^{3} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -128910779844.2735 \lor \neg \left(x \le 407.785094800499166\right):\\
\;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double code(double x) {
	return ((double) (x / ((double) (((double) (x * x)) + 1.0))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -128910779844.2735) || !(x <= 407.78509480049917))) {
		VAR = ((double) (((double) (1.0 / x)) - ((double) (((double) (1.0 / ((double) pow(x, 3.0)))) - ((double) (1.0 * ((double) (1.0 / ((double) pow(x, 5.0))))))))));
	} else {
		VAR = ((double) (((double) (x / ((double) (((double) pow(((double) (x * x)), 3.0)) + ((double) pow(1.0, 3.0)))))) * ((double) (((double) (((double) (x * x)) * ((double) (x * x)))) + ((double) (((double) (1.0 * 1.0)) - ((double) (((double) (x * x)) * 1.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -128910779844.2735 or 407.785094800499166 < x

    1. Initial program 31.3

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

    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

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

    if -128910779844.2735 < x < 407.785094800499166

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

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

    4. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -128910779844.2735 \lor \neg \left(x \le 407.785094800499166\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\left(x \cdot x\right)}^{3} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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