Average Error: 15.1 → 0.1
Time: 13.8s
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 r3364356 = x;
        double r3364357 = r3364356 * r3364356;
        double r3364358 = 1.0;
        double r3364359 = r3364357 + r3364358;
        double r3364360 = r3364356 / r3364359;
        return r3364360;
}
↓
double f(double x) {
        double r3364361 = 1.0;
        double r3364362 = x;
        double r3364363 = 1.0;
        double r3364364 = r3364363 / r3364362;
        double r3364365 = r3364362 + r3364364;
        double r3364366 = r3364361 / r3364365;
        return r3364366;
}

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

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

  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 2019169 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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