x / (x^2 + 1)

Average Error: 14.9 → 0.2

Time: 5.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0108746280769982 \lor \neg \left(x \le 1.0073267373299459\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + {x}^{5}\right) - {x}^{3}\right)\\ \end{array}\]

C

double f(double x) {
        double r57687 = x;
        double r57688 = r57687 * r57687;
        double r57689 = 1.0;
        double r57690 = r57688 + r57689;
        double r57691 = r57687 / r57690;
        return r57691;
}

↓

double f(double x) {
        double r57692 = x;
        double r57693 = -1.0108746280769982;
        bool r57694 = r57692 <= r57693;
        double r57695 = 1.007326737329946;
        bool r57696 = r57692 <= r57695;
        double r57697 = !r57696;
        bool r57698 = r57694 || r57697;
        double r57699 = 1.0;
        double r57700 = 5.0;
        double r57701 = pow(r57692, r57700);
        double r57702 = r57699 / r57701;
        double r57703 = 1.0;
        double r57704 = r57703 / r57692;
        double r57705 = r57702 + r57704;
        double r57706 = 3.0;
        double r57707 = pow(r57692, r57706);
        double r57708 = r57699 / r57707;
        double r57709 = r57705 - r57708;
        double r57710 = r57692 + r57701;
        double r57711 = r57710 - r57707;
        double r57712 = r57699 * r57711;
        double r57713 = r57698 ? r57709 : r57712;
        return r57713;
}

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0108746280769982 or 1.007326737329946 < x

   1. Initial program 29.3

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

   if -1.0108746280769982 < x < 1.007326737329946

   1. Initial program 0.0

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

      \[\leadsto \color{blue}{1 \cdot \left(\left(x + {x}^{5}\right) - {x}^{3}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0108746280769982 \lor \neg \left(x \le 1.0073267373299459\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + {x}^{5}\right) - {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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