Average Error: 15.1 → 0.1
Time: 40.7s
Precision: 64
\[\frac{x}{x \cdot x + 1.0}\]
↓
\[\frac{1}{x + \frac{1.0}{x}}\]
\frac{x}{x \cdot x + 1.0}
↓
\frac{1}{x + \frac{1.0}{x}}
double f(double x) {
        double r3490355 = x;
        double r3490356 = r3490355 * r3490355;
        double r3490357 = 1.0;
        double r3490358 = r3490356 + r3490357;
        double r3490359 = r3490355 / r3490358;
        return r3490359;
}
↓
double f(double x) {
        double r3490360 = 1.0;
        double r3490361 = x;
        double r3490362 = 1.0;
        double r3490363 = r3490362 / r3490361;
        double r3490364 = r3490361 + r3490363;
        double r3490365 = r3490360 / r3490364;
        return r3490365;
}

Error

Bits error versus x

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Results

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Target

Original15.1
Target0.1
Herbie0.1
\[\frac{1.0}{x + \frac{1.0}{x}}\]

Derivation

  1. Initial program 15.1

    \[\frac{x}{x \cdot x + 1.0}\]
  2. Using strategy rm

  3. Applied clear-num15.1

    \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1.0}{x}}}\]

  4. Taylor expanded around 0 0.1

    \[\leadsto \frac{1}{\color{blue}{x + 1.0 \cdot \frac{1}{x}}}\]

  5. Simplified0.1

    \[\leadsto \frac{1}{\color{blue}{x + \frac{1.0}{x}}}\]

  6. Final simplification0.1

    \[\leadsto \frac{1}{x + \frac{1.0}{x}}\]

Reproduce

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

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))