Average Error: 14.4 → 0.1
Time: 6.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{\frac{1}{x} + x}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{\frac{1}{x} + x}
double f(double x) {
        double r54928 = x;
        double r54929 = r54928 * r54928;
        double r54930 = 1.0;
        double r54931 = r54929 + r54930;
        double r54932 = r54928 / r54931;
        return r54932;
}
↓
double f(double x) {
        double r54933 = 1.0;
        double r54934 = 1.0;
        double r54935 = x;
        double r54936 = r54934 / r54935;
        double r54937 = r54936 + r54935;
        double r54938 = r54933 / r54937;
        return r54938;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.4
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.4

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

  3. Applied clear-num14.4

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

  4. Taylor expanded around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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