Average Error: 15.0 → 0.0
Time: 3.9s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\

\end{array}
double code(double x) {
	return (x / ((x * x) + 1.0));
}
↓
double code(double x) {
	double temp;
	if (((x <= -14917877722.165144) || !(x <= 764.6675213818143))) {
		temp = fma(1.0, ((1.0 / pow(x, 5.0)) - (1.0 / pow(x, 3.0))), (1.0 / x));
	} else {
		temp = ((x / sqrt(((x * x) + 1.0))) / (hypot(pow(1.0, 1.5), pow(x, 3.0)) / sqrt((((x * x) * (x * x)) + ((1.0 * 1.0) - ((x * x) * 1.0))))));
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -14917877722.165144 or 764.6675213818143 < x

    1. Initial program 30.2

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

    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

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

    if -14917877722.165144 < x < 764.6675213818143

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
    5. Using strategy rm

    6. Applied flip3-+0.0

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

    7. Applied sqrt-div0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14917877722.165144 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\frac{\mathsf{hypot}\left({1}^{\frac{3}{2}}, {x}^{3}\right)}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))