Average Error: 10.1 → 0.1
Time: 15.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x - 1}}{\left(x + 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x - 1}}{\left(x + 1\right) \cdot x}
double f(double x) {
        double r5660596 = 1.0;
        double r5660597 = x;
        double r5660598 = r5660597 + r5660596;
        double r5660599 = r5660596 / r5660598;
        double r5660600 = 2.0;
        double r5660601 = r5660600 / r5660597;
        double r5660602 = r5660599 - r5660601;
        double r5660603 = r5660597 - r5660596;
        double r5660604 = r5660596 / r5660603;
        double r5660605 = r5660602 + r5660604;
        return r5660605;
}
↓
double f(double x) {
        double r5660606 = 2.0;
        double r5660607 = x;
        double r5660608 = 1.0;
        double r5660609 = r5660607 - r5660608;
        double r5660610 = r5660606 / r5660609;
        double r5660611 = r5660607 + r5660608;
        double r5660612 = r5660611 * r5660607;
        double r5660613 = r5660610 / r5660612;
        return r5660613;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.1
Target0.2
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-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-add26.0

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

  5. Taylor expanded around 0 0.2

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

  7. Applied *-un-lft-identity0.2

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

  8. Applied times-frac0.1

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

  10. Applied associate-*l/0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 
(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))))