Average Error: 9.9 → 0.1
Time: 20.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}
double f(double x) {
        double r4004491 = 1.0;
        double r4004492 = x;
        double r4004493 = r4004492 + r4004491;
        double r4004494 = r4004491 / r4004493;
        double r4004495 = 2.0;
        double r4004496 = r4004495 / r4004492;
        double r4004497 = r4004494 - r4004496;
        double r4004498 = r4004492 - r4004491;
        double r4004499 = r4004491 / r4004498;
        double r4004500 = r4004497 + r4004499;
        return r4004500;
}
↓
double f(double x) {
        double r4004501 = 2.0;
        double r4004502 = x;
        double r4004503 = 1.0;
        double r4004504 = r4004502 + r4004503;
        double r4004505 = r4004501 / r4004504;
        double r4004506 = r4004505 / r4004502;
        double r4004507 = r4004502 - r4004503;
        double r4004508 = r4004506 / r4004507;
        return r4004508;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

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

  3. Applied frac-sub25.4

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

  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

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

  6. Taylor expanded around 0 0.3

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

  8. Applied associate-/r*0.1

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

  10. Applied associate-/r*0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{x + 1}}{x}}}{x - 1}\]

  11. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]

Reproduce

herbie shell --seed 2019172 +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))))