Average Error: 9.8 → 0.3
Time: 13.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - 1 \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - 1 \cdot x}
double f(double x) {
        double r87540 = 1.0;
        double r87541 = x;
        double r87542 = r87541 + r87540;
        double r87543 = r87540 / r87542;
        double r87544 = 2.0;
        double r87545 = r87544 / r87541;
        double r87546 = r87543 - r87545;
        double r87547 = r87541 - r87540;
        double r87548 = r87540 / r87547;
        double r87549 = r87546 + r87548;
        return r87549;
}
↓
double f(double x) {
        double r87550 = 2.0;
        double r87551 = x;
        double r87552 = 3.0;
        double r87553 = pow(r87551, r87552);
        double r87554 = 1.0;
        double r87555 = r87554 * r87551;
        double r87556 = r87553 - r87555;
        double r87557 = r87550 / r87556;
        return r87557;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

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 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-add25.9

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

  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. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - 1 \cdot x}}\]

  7. Final simplification0.3

    \[\leadsto \frac{2}{{x}^{3} - 1 \cdot x}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))