Average Error: 14.6 → 0.1
Time: 2.0s
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 r51098 = x;
        double r51099 = r51098 * r51098;
        double r51100 = 1.0;
        double r51101 = r51099 + r51100;
        double r51102 = r51098 / r51101;
        return r51102;
}
↓
double f(double x) {
        double r51103 = 1.0;
        double r51104 = x;
        double r51105 = 1.0;
        double r51106 = r51105 / r51104;
        double r51107 = r51104 + r51106;
        double r51108 = r51103 / r51107;
        return r51108;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original14.6
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.6

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

  3. Applied clear-num14.6

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

  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 2020035 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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