x / (x^2 + 1)

Average Error: 14.7 → 0.0

Time: 15.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13757.725196327832:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 353.58163347243044:\\ \;\;\;\;\frac{x}{\left|\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}\right|} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r2489546 = x;
        double r2489547 = r2489546 * r2489546;
        double r2489548 = 1.0;
        double r2489549 = r2489547 + r2489548;
        double r2489550 = r2489546 / r2489549;
        return r2489550;
}

↓

double f(double x) {
        double r2489551 = x;
        double r2489552 = -13757.725196327832;
        bool r2489553 = r2489551 <= r2489552;
        double r2489554 = 1.0;
        double r2489555 = r2489554 / r2489551;
        double r2489556 = r2489551 * r2489551;
        double r2489557 = r2489555 / r2489556;
        double r2489558 = r2489555 - r2489557;
        double r2489559 = 5.0;
        double r2489560 = pow(r2489551, r2489559);
        double r2489561 = r2489554 / r2489560;
        double r2489562 = r2489558 + r2489561;
        double r2489563 = 353.58163347243044;
        bool r2489564 = r2489551 <= r2489563;
        double r2489565 = fma(r2489551, r2489551, r2489554);
        double r2489566 = cbrt(r2489565);
        double r2489567 = fabs(r2489566);
        double r2489568 = r2489551 / r2489567;
        double r2489569 = sqrt(r2489565);
        double r2489570 = r2489554 / r2489569;
        double r2489571 = sqrt(r2489566);
        double r2489572 = r2489570 / r2489571;
        double r2489573 = r2489568 * r2489572;
        double r2489574 = r2489564 ? r2489573 : r2489562;
        double r2489575 = r2489553 ? r2489562 : r2489574;
        return r2489575;
}

Error

Bits error versus x

Target

Original	14.7
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -13757.725196327832 or 353.58163347243044 < x

   1. Initial program 29.8

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

   2. Simplified 29.8

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 29.8

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

   5. Applied associate-/r* 29.7

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

   6. Taylor expanded around inf 0.0

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

   7. Simplified 0.0

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

   if -13757.725196327832 < x < 353.58163347243044

   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. Using strategy rm

   4. Applied add-sqr-sqrt 0.0

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

   5. Applied associate-/r* 0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.0

      \[\leadsto \frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}}\]

   8. Applied sqrt-prod 0.0

      \[\leadsto \frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\color{blue}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}}\]

   9. Applied div-inv 0.0

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}\]

   10. Applied times-frac 0.0

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}}\]

   11. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13757.725196327832:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 353.58163347243044:\\ \;\;\;\;\frac{x}{\left|\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}\right|} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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