Average Error: 9.9 → 0.1
Time: 14.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r5115410 = 1.0;
        double r5115411 = x;
        double r5115412 = r5115411 + r5115410;
        double r5115413 = r5115410 / r5115412;
        double r5115414 = 2.0;
        double r5115415 = r5115414 / r5115411;
        double r5115416 = r5115413 - r5115415;
        double r5115417 = r5115411 - r5115410;
        double r5115418 = r5115410 / r5115417;
        double r5115419 = r5115416 + r5115418;
        return r5115419;
}
↓
double f(double x) {
        double r5115420 = 2.0;
        double r5115421 = x;
        double r5115422 = 1.0;
        double r5115423 = r5115421 + r5115422;
        double r5115424 = r5115423 * r5115421;
        double r5115425 = r5115420 / r5115424;
        double r5115426 = r5115421 - r5115422;
        double r5115427 = r5115425 / r5115426;
        return r5115427;
}

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. Taylor expanded around 0 0.3

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 
(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))))