Average Error: 9.7 → 0.3
Time: 14.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}
double f(double x) {
        double r2948186 = 1.0;
        double r2948187 = x;
        double r2948188 = r2948187 + r2948186;
        double r2948189 = r2948186 / r2948188;
        double r2948190 = 2.0;
        double r2948191 = r2948190 / r2948187;
        double r2948192 = r2948189 - r2948191;
        double r2948193 = r2948187 - r2948186;
        double r2948194 = r2948186 / r2948193;
        double r2948195 = r2948192 + r2948194;
        return r2948195;
}
↓
double f(double x) {
        double r2948196 = 2.0;
        double r2948197 = x;
        double r2948198 = 1.0;
        double r2948199 = r2948197 - r2948198;
        double r2948200 = r2948197 + r2948198;
        double r2948201 = r2948197 * r2948200;
        double r2948202 = r2948199 * r2948201;
        double r2948203 = r2948196 / r2948202;
        return r2948203;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied +-commutative9.7

    \[\leadsto \color{blue}{\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)}\]
  4. Using strategy rm

  5. Applied frac-sub26.3

    \[\leadsto \frac{1}{x - 1} + \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}}\]

  6. Applied frac-add25.6

    \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(x + 1\right) \cdot x\right) + \left(x - 1\right) \cdot \left(1 \cdot x - \left(x + 1\right) \cdot 2\right)}{\left(x - 1\right) \cdot \left(\left(x + 1\right) \cdot x\right)}}\]

  7. Simplified25.6

    \[\leadsto \frac{\color{blue}{\left(x - 1\right) \cdot \left(x + -2 \cdot \left(x + 1\right)\right) + \left(x + 1\right) \cdot x}}{\left(x - 1\right) \cdot \left(\left(x + 1\right) \cdot x\right)}\]

  8. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(x - 1\right) \cdot \left(\left(x + 1\right) \cdot x\right)}\]

  9. Final simplification0.3

    \[\leadsto \frac{2}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))