Average Error: 9.8 → 0.1
Time: 2.9m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{(x \cdot x + -1)_*}}{x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{(x \cdot x + -1)_*}}{x}
double f(double x) {
        double r21787451 = 1.0;
        double r21787452 = x;
        double r21787453 = r21787452 + r21787451;
        double r21787454 = r21787451 / r21787453;
        double r21787455 = 2.0;
        double r21787456 = r21787455 / r21787452;
        double r21787457 = r21787454 - r21787456;
        double r21787458 = r21787452 - r21787451;
        double r21787459 = r21787451 / r21787458;
        double r21787460 = r21787457 + r21787459;
        return r21787460;
}
↓
double f(double x) {
        double r21787461 = 2.0;
        double r21787462 = x;
        double r21787463 = -1.0;
        double r21787464 = fma(r21787462, r21787462, r21787463);
        double r21787465 = r21787461 / r21787464;
        double r21787466 = r21787465 / r21787462;
        return r21787466;
}

Error

Bits error versus x

Target

Original9.8
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.8

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

  3. Applied frac-sub25.8

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

  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

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

  6. Simplified25.4

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

    \[\leadsto \frac{\frac{2}{(x \cdot x + -1)_*}}{x}\]

Reproduce

herbie shell --seed 2019112 +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))))