Average Error: 14.7 → 0.1
Time: 1.0s
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 r74358 = x;
        double r74359 = r74358 * r74358;
        double r74360 = 1.0;
        double r74361 = r74359 + r74360;
        double r74362 = r74358 / r74361;
        return r74362;
}
↓
double f(double x) {
        double r74363 = 1.0;
        double r74364 = x;
        double r74365 = 1.0;
        double r74366 = r74365 / r74364;
        double r74367 = r74364 + r74366;
        double r74368 = r74363 / r74367;
        return r74368;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.7

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

  3. Applied clear-num14.7

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

  4. Taylor expanded around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020003 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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