x / (x^2 + 1)

Average Error: 15.1 → 0.0

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r915662 = x;
        double r915663 = r915662 * r915662;
        double r915664 = 1.0;
        double r915665 = r915663 + r915664;
        double r915666 = r915662 / r915665;
        return r915666;
}

↓

double f(double x) {
        double r915667 = x;
        double r915668 = -2678435.1094305497;
        bool r915669 = r915667 <= r915668;
        double r915670 = 1.0;
        double r915671 = r915670 / r915667;
        double r915672 = r915667 * r915667;
        double r915673 = r915671 / r915672;
        double r915674 = r915671 - r915673;
        double r915675 = 5.0;
        double r915676 = pow(r915667, r915675);
        double r915677 = r915670 / r915676;
        double r915678 = r915674 + r915677;
        double r915679 = 11907527.731338572;
        bool r915680 = r915667 <= r915679;
        double r915681 = fma(r915667, r915667, r915670);
        double r915682 = r915667 / r915681;
        double r915683 = r915680 ? r915682 : r915678;
        double r915684 = r915669 ? r915678 : r915683;
        return r915684;
}

Error

Bits error versus x

Target

Original	15.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -2678435.1094305497 or 11907527.731338572 < x

   1. Initial program 30.4

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

   2. Simplified 30.4

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -2678435.1094305497 < x < 11907527.731338572

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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