Average Error: 15.0 → 0.1
Time: 15.1s
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 r41065 = x;
        double r41066 = r41065 * r41065;
        double r41067 = 1.0;
        double r41068 = r41066 + r41067;
        double r41069 = r41065 / r41068;
        return r41069;
}
↓
double f(double x) {
        double r41070 = 1.0;
        double r41071 = 1.0;
        double r41072 = x;
        double r41073 = r41071 / r41072;
        double r41074 = r41073 + r41072;
        double r41075 = r41070 / r41074;
        return r41075;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 15.0

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

  3. Applied clear-num15.0

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

  4. Simplified15.0

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

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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