x / (x^2 + 1)

Average Error: 14.9 → 0.0

Time: 21.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.182351251673564960881259272718336657084 \cdot 10^{51}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 268164.477104543824680149555206298828125:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \end{array}\]

C

double f(double x) {
        double r2809559 = x;
        double r2809560 = r2809559 * r2809559;
        double r2809561 = 1.0;
        double r2809562 = r2809560 + r2809561;
        double r2809563 = r2809559 / r2809562;
        return r2809563;
}

↓

double f(double x) {
        double r2809564 = x;
        double r2809565 = -2.182351251673565e+51;
        bool r2809566 = r2809564 <= r2809565;
        double r2809567 = 1.0;
        double r2809568 = r2809567 / r2809564;
        double r2809569 = 1.0;
        double r2809570 = r2809564 * r2809564;
        double r2809571 = r2809570 * r2809564;
        double r2809572 = r2809569 / r2809571;
        double r2809573 = 5.0;
        double r2809574 = pow(r2809564, r2809573);
        double r2809575 = r2809569 / r2809574;
        double r2809576 = r2809572 - r2809575;
        double r2809577 = r2809568 - r2809576;
        double r2809578 = 268164.4771045438;
        bool r2809579 = r2809564 <= r2809578;
        double r2809580 = fma(r2809564, r2809564, r2809569);
        double r2809581 = r2809564 / r2809580;
        double r2809582 = r2809579 ? r2809581 : r2809577;
        double r2809583 = r2809566 ? r2809577 : r2809582;
        return r2809583;
}

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -2.182351251673565e+51 or 268164.4771045438 < x

   1. Initial program 32.5

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

   2. Simplified 32.5

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -2.182351251673565e+51 < x < 268164.4771045438

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.182351251673564960881259272718336657084 \cdot 10^{51}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 268164.477104543824680149555206298828125:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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