Average Error: 10.1 → 0.1
Time: 23.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[2 \cdot \frac{\frac{1}{x}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{\frac{1}{x}}{\left(x + 1\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r97497 = 1.0;
        double r97498 = x;
        double r97499 = r97498 + r97497;
        double r97500 = r97497 / r97499;
        double r97501 = 2.0;
        double r97502 = r97501 / r97498;
        double r97503 = r97500 - r97502;
        double r97504 = r97498 - r97497;
        double r97505 = r97497 / r97504;
        double r97506 = r97503 + r97505;
        return r97506;
}
↓
double f(double x) {
        double r97507 = 2.0;
        double r97508 = 1.0;
        double r97509 = x;
        double r97510 = r97508 / r97509;
        double r97511 = 1.0;
        double r97512 = r97509 + r97511;
        double r97513 = r97509 - r97511;
        double r97514 = r97512 * r97513;
        double r97515 = r97510 / r97514;
        double r97516 = r97507 * r97515;
        return r97516;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sub25.9

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

  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. Simplified25.4

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

  6. Simplified25.4

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

  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 div-inv0.3

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))