Average Error: 9.8 → 0.1
Time: 37.8s
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 r3953391 = 1.0;
        double r3953392 = x;
        double r3953393 = r3953392 + r3953391;
        double r3953394 = r3953391 / r3953393;
        double r3953395 = 2.0;
        double r3953396 = r3953395 / r3953392;
        double r3953397 = r3953394 - r3953396;
        double r3953398 = r3953392 - r3953391;
        double r3953399 = r3953391 / r3953398;
        double r3953400 = r3953397 + r3953399;
        return r3953400;
}
↓
double f(double x) {
        double r3953401 = 2.0;
        double r3953402 = x;
        double r3953403 = fma(r3953402, r3953402, r3953402);
        double r3953404 = r3953401 / r3953403;
        double r3953405 = 1.0;
        double r3953406 = r3953402 - r3953405;
        double r3953407 = r3953404 / r3953406;
        return r3953407;
}

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 *-un-lft-identity9.8

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

  4. Applied associate-/l*9.8

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

  5. Simplified9.8

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

  7. Applied frac-sub26.0

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

  8. Applied frac-add25.4

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

  9. Taylor expanded around -inf 0.3

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

  11. Applied associate-/r*0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019151 +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))))