Average Error: 10.1 → 0.1
Time: 18.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1}
double f(double x) {
        double r4461692 = 1.0;
        double r4461693 = x;
        double r4461694 = r4461693 + r4461692;
        double r4461695 = r4461692 / r4461694;
        double r4461696 = 2.0;
        double r4461697 = r4461696 / r4461693;
        double r4461698 = r4461695 - r4461697;
        double r4461699 = r4461693 - r4461692;
        double r4461700 = r4461692 / r4461699;
        double r4461701 = r4461698 + r4461700;
        return r4461701;
}
↓
double f(double x) {
        double r4461702 = 2.0;
        double r4461703 = x;
        double r4461704 = fma(r4461703, r4461703, r4461703);
        double r4461705 = r4461702 / r4461704;
        double r4461706 = 1.0;
        double r4461707 = r4461703 - r4461706;
        double r4461708 = r4461705 / r4461707;
        return r4461708;
}

Error

Bits error versus x

Target

Original10.1
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.1

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

  3. Applied frac-sub25.7

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

  4. Applied frac-add25.1

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

  5. Simplified25.5

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

  6. Simplified25.5

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 +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))))