Average Error: 9.7 → 0.1
Time: 18.4s
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 r4574049 = 1.0;
        double r4574050 = x;
        double r4574051 = r4574050 + r4574049;
        double r4574052 = r4574049 / r4574051;
        double r4574053 = 2.0;
        double r4574054 = r4574053 / r4574050;
        double r4574055 = r4574052 - r4574054;
        double r4574056 = r4574050 - r4574049;
        double r4574057 = r4574049 / r4574056;
        double r4574058 = r4574055 + r4574057;
        return r4574058;
}
↓
double f(double x) {
        double r4574059 = 2.0;
        double r4574060 = x;
        double r4574061 = fma(r4574060, r4574060, r4574060);
        double r4574062 = r4574059 / r4574061;
        double r4574063 = 1.0;
        double r4574064 = r4574060 - r4574063;
        double r4574065 = r4574062 / r4574064;
        return r4574065;
}

Error

Bits error versus x

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub26.3

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

  4. Applied frac-add25.6

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

  5. Simplified25.9

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

  6. Simplified25.9

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  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 2019163 +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))))