x / (x^2 + 1)

Average Error: 14.6 → 0.0

Time: 7.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.333300421862763794492150706903049815887 \cdot 10^{154} \lor \neg \left(x \le 982.8796450984181092280778102576732635498\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \end{array}\]

C

double f(double x) {
        double r112106 = x;
        double r112107 = r112106 * r112106;
        double r112108 = 1.0;
        double r112109 = r112107 + r112108;
        double r112110 = r112106 / r112109;
        return r112110;
}

↓

double f(double x) {
        double r112111 = x;
        double r112112 = -1.3333004218627638e+154;
        bool r112113 = r112111 <= r112112;
        double r112114 = 982.8796450984181;
        bool r112115 = r112111 <= r112114;
        double r112116 = !r112115;
        bool r112117 = r112113 || r112116;
        double r112118 = 1.0;
        double r112119 = 5.0;
        double r112120 = pow(r112111, r112119);
        double r112121 = r112118 / r112120;
        double r112122 = 1.0;
        double r112123 = r112122 / r112111;
        double r112124 = 3.0;
        double r112125 = pow(r112111, r112124);
        double r112126 = r112118 / r112125;
        double r112127 = r112123 - r112126;
        double r112128 = r112121 + r112127;
        double r112129 = fma(r112111, r112111, r112118);
        double r112130 = sqrt(r112129);
        double r112131 = r112122 / r112130;
        double r112132 = r112111 / r112130;
        double r112133 = r112131 * r112132;
        double r112134 = r112117 ? r112128 : r112133;
        return r112134;
}

Error

Bits error versus x

Target

Original	14.6
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3333004218627638e+154 or 982.8796450984181 < x

   1. Initial program 39.3

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

   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. Simplified 0.0

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

   if -1.3333004218627638e+154 < x < 982.8796450984181

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied *-un-lft-identity 0.1

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

   5. Applied times-frac 0.0

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

   6. Simplified 0.0

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

   7. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.333300421862763794492150706903049815887 \cdot 10^{154} \lor \neg \left(x \le 982.8796450984181092280778102576732635498\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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