x / (x^2 + 1)

Average Error: 14.7 → 0.2

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0028494934768706:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1.0197885553055124:\\ \;\;\;\;\left(x - \left(x \cdot x\right) \cdot x\right) + {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r2843099 = x;
        double r2843100 = r2843099 * r2843099;
        double r2843101 = 1.0;
        double r2843102 = r2843100 + r2843101;
        double r2843103 = r2843099 / r2843102;
        return r2843103;
}

↓

double f(double x) {
        double r2843104 = x;
        double r2843105 = -1.0028494934768706;
        bool r2843106 = r2843104 <= r2843105;
        double r2843107 = 1.0;
        double r2843108 = r2843107 / r2843104;
        double r2843109 = r2843104 * r2843104;
        double r2843110 = r2843109 * r2843104;
        double r2843111 = r2843107 / r2843110;
        double r2843112 = r2843108 - r2843111;
        double r2843113 = 5.0;
        double r2843114 = pow(r2843104, r2843113);
        double r2843115 = r2843107 / r2843114;
        double r2843116 = r2843112 + r2843115;
        double r2843117 = 1.0197885553055124;
        bool r2843118 = r2843104 <= r2843117;
        double r2843119 = r2843104 - r2843110;
        double r2843120 = r2843119 + r2843114;
        double r2843121 = r2843118 ? r2843120 : r2843116;
        double r2843122 = r2843106 ? r2843116 : r2843121;
        return r2843122;
}

Error

Bits error versus x

Target

Original	14.7
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0028494934768706 or 1.0197885553055124 < x

   1. Initial program 29.3

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

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

   if -1.0028494934768706 < x < 1.0197885553055124

   1. Initial program 0.0

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0028494934768706:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1.0197885553055124:\\ \;\;\;\;\left(x - \left(x \cdot x\right) \cdot x\right) + {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

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

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

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