Average Error: 15.1 → 0.1
Time: 12.2s
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 r2772305 = x;
        double r2772306 = r2772305 * r2772305;
        double r2772307 = 1.0;
        double r2772308 = r2772306 + r2772307;
        double r2772309 = r2772305 / r2772308;
        return r2772309;
}
↓
double f(double x) {
        double r2772310 = 1.0;
        double r2772311 = x;
        double r2772312 = 1.0;
        double r2772313 = r2772312 / r2772311;
        double r2772314 = r2772311 + r2772313;
        double r2772315 = r2772310 / r2772314;
        return r2772315;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified15.1

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
  3. Using strategy rm

  4. Applied clear-num15.1

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}\]

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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