Average Error: 9.9 → 0.1
Time: 25.2s
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 r5925688 = 1.0;
        double r5925689 = x;
        double r5925690 = r5925689 + r5925688;
        double r5925691 = r5925688 / r5925690;
        double r5925692 = 2.0;
        double r5925693 = r5925692 / r5925689;
        double r5925694 = r5925691 - r5925693;
        double r5925695 = r5925689 - r5925688;
        double r5925696 = r5925688 / r5925695;
        double r5925697 = r5925694 + r5925696;
        return r5925697;
}
↓
double f(double x) {
        double r5925698 = 2.0;
        double r5925699 = x;
        double r5925700 = 1.0;
        double r5925701 = r5925699 + r5925700;
        double r5925702 = r5925701 * r5925699;
        double r5925703 = r5925698 / r5925702;
        double r5925704 = r5925699 - r5925700;
        double r5925705 = r5925703 / r5925704;
        return r5925705;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

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

  3. Applied frac-sub25.4

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

  4. Applied frac-add25.0

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

  5. Simplified25.4

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

  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 2019172 +o rules:numerics
(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))))