Average Error: 15.1 → 0.1
Time: 7.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{x + \frac{1}{x}}
double f(double x) {
        double r2665400 = x;
        double r2665401 = r2665400 * r2665400;
        double r2665402 = 1.0;
        double r2665403 = r2665401 + r2665402;
        double r2665404 = r2665400 / r2665403;
        return r2665404;
}
↓
double f(double x) {
        double r2665405 = 1.0;
        double r2665406 = x;
        double r2665407 = r2665405 / r2665406;
        double r2665408 = r2665406 + r2665407;
        double r2665409 = r2665405 / r2665408;
        return r2665409;
}

Error

Bits error versus x

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Results

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Target

Original15.1
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 15.1

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

  3. Applied clear-num15.1

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

  4. Taylor expanded around 0 0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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