Average Error: 9.9 → 0.1
Time: 1.2m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
double f(double x) {
        double r10448614 = 1.0;
        double r10448615 = x;
        double r10448616 = r10448615 + r10448614;
        double r10448617 = r10448614 / r10448616;
        double r10448618 = 2.0;
        double r10448619 = r10448618 / r10448615;
        double r10448620 = r10448617 - r10448619;
        double r10448621 = r10448615 - r10448614;
        double r10448622 = r10448614 / r10448621;
        double r10448623 = r10448620 + r10448622;
        return r10448623;
}
↓
double f(double x) {
        double r10448624 = 2.0;
        double r10448625 = x;
        double r10448626 = 1.0;
        double r10448627 = r10448625 + r10448626;
        double r10448628 = r10448627 * r10448625;
        double r10448629 = r10448624 / r10448628;
        double r10448630 = r10448625 - r10448626;
        double r10448631 = r10448629 / r10448630;
        return r10448631;
}


\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}

Error

Bits error versus x

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-sub26.0

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

  4. Applied frac-add25.4

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

  5. Taylor expanded around inf 0.3

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019102 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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