x / (x^2 + 1)

Average Error: 15.1 → 0.0

Time: 8.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 2168.52278018713832\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

C

double f(double x) {
        double r54343 = x;
        double r54344 = r54343 * r54343;
        double r54345 = 1.0;
        double r54346 = r54344 + r54345;
        double r54347 = r54343 / r54346;
        return r54347;
}

↓

double f(double x) {
        double r54348 = x;
        double r54349 = -62379821099783.21;
        bool r54350 = r54348 <= r54349;
        double r54351 = 2168.5227801871383;
        bool r54352 = r54348 <= r54351;
        double r54353 = !r54352;
        bool r54354 = r54350 || r54353;
        double r54355 = 1.0;
        double r54356 = r54355 / r54348;
        double r54357 = 1.0;
        double r54358 = 5.0;
        double r54359 = pow(r54348, r54358);
        double r54360 = r54357 / r54359;
        double r54361 = r54356 + r54360;
        double r54362 = 3.0;
        double r54363 = pow(r54348, r54362);
        double r54364 = r54357 / r54363;
        double r54365 = r54361 - r54364;
        double r54366 = fma(r54348, r54348, r54357);
        double r54367 = r54355 / r54366;
        double r54368 = r54348 * r54367;
        double r54369 = r54354 ? r54365 : r54368;
        return r54369;
}

Error

Bits error versus x

Target

Original	15.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -62379821099783.21 or 2168.5227801871383 < x

   1. Initial program 30.9

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

   2. Simplified 30.9

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

   4. Applied *-un-lft-identity 30.9

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

   5. Applied add-cube-cbrt 31.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot \mathsf{fma}\left(x, x, 1\right)}\]

   6. Applied times-frac 31.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{\mathsf{fma}\left(x, x, 1\right)}}\]

   7. Simplified 31.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{\mathsf{fma}\left(x, x, 1\right)}\]

   8. Taylor expanded around inf 0.0

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

   9. Simplified 0.0

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

   if -62379821099783.21 < x < 2168.5227801871383

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 2168.52278018713832\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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