Average Error: 9.9 → 0.3
Time: 29.6s
Precision: 64
\[\left(\frac{1.0}{x + 1.0} - \frac{2.0}{x}\right) + \frac{1.0}{x - 1.0}\]
↓
\[\frac{2.0}{\left(\left(x + 1.0\right) \cdot x\right) \cdot \left(x - 1.0\right)}\]
\left(\frac{1.0}{x + 1.0} - \frac{2.0}{x}\right) + \frac{1.0}{x - 1.0}
↓
\frac{2.0}{\left(\left(x + 1.0\right) \cdot x\right) \cdot \left(x - 1.0\right)}
double f(double x) {
        double r6121141 = 1.0;
        double r6121142 = x;
        double r6121143 = r6121142 + r6121141;
        double r6121144 = r6121141 / r6121143;
        double r6121145 = 2.0;
        double r6121146 = r6121145 / r6121142;
        double r6121147 = r6121144 - r6121146;
        double r6121148 = r6121142 - r6121141;
        double r6121149 = r6121141 / r6121148;
        double r6121150 = r6121147 + r6121149;
        return r6121150;
}
↓
double f(double x) {
        double r6121151 = 2.0;
        double r6121152 = x;
        double r6121153 = 1.0;
        double r6121154 = r6121152 + r6121153;
        double r6121155 = r6121154 * r6121152;
        double r6121156 = r6121152 - r6121153;
        double r6121157 = r6121155 * r6121156;
        double r6121158 = r6121151 / r6121157;
        return r6121158;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.3
\[\frac{2.0}{x \cdot \left(x \cdot x - 1.0\right)}\]

Derivation

  1. Initial program 9.9

    \[\left(\frac{1.0}{x + 1.0} - \frac{2.0}{x}\right) + \frac{1.0}{x - 1.0}\]
  2. Using strategy rm

  3. Applied frac-sub26.3

    \[\leadsto \color{blue}{\frac{1.0 \cdot x - \left(x + 1.0\right) \cdot 2.0}{\left(x + 1.0\right) \cdot x}} + \frac{1.0}{x - 1.0}\]

  4. Applied frac-add25.7

    \[\leadsto \color{blue}{\frac{\left(1.0 \cdot x - \left(x + 1.0\right) \cdot 2.0\right) \cdot \left(x - 1.0\right) + \left(\left(x + 1.0\right) \cdot x\right) \cdot 1.0}{\left(\left(x + 1.0\right) \cdot x\right) \cdot \left(x - 1.0\right)}}\]

  5. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2.0}}{\left(\left(x + 1.0\right) \cdot x\right) \cdot \left(x - 1.0\right)}\]

  6. Final simplification0.3

    \[\leadsto \frac{2.0}{\left(\left(x + 1.0\right) \cdot x\right) \cdot \left(x - 1.0\right)}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))