x / (x^2 + 1)

Average Error: 14.8 → 0.0

Time: 11.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -90716742906700070912 \lor \neg \left(x \le 8839590.60347673483192920684814453125\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

C

double f(double x) {
        double r44105 = x;
        double r44106 = r44105 * r44105;
        double r44107 = 1.0;
        double r44108 = r44106 + r44107;
        double r44109 = r44105 / r44108;
        return r44109;
}

↓

double f(double x) {
        double r44110 = x;
        double r44111 = -9.071674290670007e+19;
        bool r44112 = r44110 <= r44111;
        double r44113 = 8839590.603476735;
        bool r44114 = r44110 <= r44113;
        double r44115 = !r44114;
        bool r44116 = r44112 || r44115;
        double r44117 = 1.0;
        double r44118 = 5.0;
        double r44119 = pow(r44110, r44118);
        double r44120 = r44117 / r44119;
        double r44121 = 1.0;
        double r44122 = r44121 / r44110;
        double r44123 = 3.0;
        double r44124 = pow(r44110, r44123);
        double r44125 = r44117 / r44124;
        double r44126 = r44122 - r44125;
        double r44127 = r44120 + r44126;
        double r44128 = fma(r44110, r44110, r44117);
        double r44129 = r44121 / r44128;
        double r44130 = r44110 * r44129;
        double r44131 = r44116 ? r44127 : r44130;
        return r44131;
}

Error

Bits error versus x

Target

Original	14.8
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -9.071674290670007e+19 or 8839590.603476735 < x

   1. Initial program 31.2

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

   2. Simplified 31.2

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

   4. Applied add-sqr-sqrt 31.3

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

   5. Applied *-un-lft-identity 31.3

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

   6. Applied times-frac 31.1

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

   7. Taylor expanded around inf 0.0

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

   8. Simplified 0.0

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

   if -9.071674290670007e+19 < x < 8839590.603476735

   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 div-inv 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -90716742906700070912 \lor \neg \left(x \le 8839590.60347673483192920684814453125\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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