Average Error: 10.1 → 0.1
Time: 7.4s
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 r65545 = 1.0;
        double r65546 = x;
        double r65547 = r65546 + r65545;
        double r65548 = r65545 / r65547;
        double r65549 = 2.0;
        double r65550 = r65549 / r65546;
        double r65551 = r65548 - r65550;
        double r65552 = r65546 - r65545;
        double r65553 = r65545 / r65552;
        double r65554 = r65551 + r65553;
        return r65554;
}
↓
double f(double x) {
        double r65555 = 2.0;
        double r65556 = x;
        double r65557 = 1.0;
        double r65558 = r65556 - r65557;
        double r65559 = r65555 / r65558;
        double r65560 = r65556 + r65557;
        double r65561 = r65560 * r65556;
        double r65562 = r65559 / r65561;
        return r65562;
}

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-sub26.1

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

  4. Applied frac-add25.7

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

  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 *-un-lft-identity0.3

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

  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 2019308 
(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))))