Average Error: 10.0 → 0.1
Time: 18.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r3968797 = 1.0;
        double r3968798 = x;
        double r3968799 = r3968798 + r3968797;
        double r3968800 = r3968797 / r3968799;
        double r3968801 = 2.0;
        double r3968802 = r3968801 / r3968798;
        double r3968803 = r3968800 - r3968802;
        double r3968804 = r3968798 - r3968797;
        double r3968805 = r3968797 / r3968804;
        double r3968806 = r3968803 + r3968805;
        return r3968806;
}
↓
double f(double x) {
        double r3968807 = 2.0;
        double r3968808 = x;
        double r3968809 = 1.0;
        double r3968810 = r3968808 + r3968809;
        double r3968811 = r3968810 * r3968808;
        double r3968812 = r3968807 / r3968811;
        double r3968813 = r3968808 - r3968809;
        double r3968814 = r3968812 / r3968813;
        return r3968814;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

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

  3. Applied frac-sub26.2

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

  4. Applied frac-add25.5

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

  5. Simplified25.8

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

  6. Taylor expanded around 0 0.3

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

  8. Applied associate-/r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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