x / (x^2 + 1)

Average Error: 14.9 → 0.3

Time: 18.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0116525285447533:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 0.9949385524453179:\\ \;\;\;\;\left({x}^{5} + x\right) - x \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

C

double f(double x) {
        double r1762559 = x;
        double r1762560 = r1762559 * r1762559;
        double r1762561 = 1.0;
        double r1762562 = r1762560 + r1762561;
        double r1762563 = r1762559 / r1762562;
        return r1762563;
}

↓

double f(double x) {
        double r1762564 = x;
        double r1762565 = -1.0116525285447533;
        bool r1762566 = r1762564 <= r1762565;
        double r1762567 = 1.0;
        double r1762568 = r1762567 / r1762564;
        double r1762569 = 5.0;
        double r1762570 = pow(r1762564, r1762569);
        double r1762571 = r1762567 / r1762570;
        double r1762572 = r1762564 * r1762564;
        double r1762573 = r1762568 / r1762572;
        double r1762574 = r1762571 - r1762573;
        double r1762575 = r1762568 + r1762574;
        double r1762576 = 0.9949385524453179;
        bool r1762577 = r1762564 <= r1762576;
        double r1762578 = r1762570 + r1762564;
        double r1762579 = r1762564 * r1762572;
        double r1762580 = r1762578 - r1762579;
        double r1762581 = r1762577 ? r1762580 : r1762575;
        double r1762582 = r1762566 ? r1762575 : r1762581;
        return r1762582;
}

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0116525285447533 or 0.9949385524453179 < x

   1. Initial program 29.3

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

   2. Simplified 29.3

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

   3. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

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

   if -1.0116525285447533 < x < 0.9949385524453179

   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 *-un-lft-identity 0.0

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

   6. Applied times-frac 0.0

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

   7. Taylor expanded around 0 0.3

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

   8. Simplified 0.3

      \[\leadsto \color{blue}{\left({x}^{5} + x\right) - x \cdot \left(x \cdot x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0116525285447533:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 0.9949385524453179:\\ \;\;\;\;\left({x}^{5} + x\right) - x \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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