Average Error: 9.4 → 0.1
Time: 49.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{x - 1} \cdot \frac{2}{\left(x + 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{x - 1} \cdot \frac{2}{\left(x + 1\right) \cdot x}
double f(double x) {
        double r3638055 = 1.0;
        double r3638056 = x;
        double r3638057 = r3638056 + r3638055;
        double r3638058 = r3638055 / r3638057;
        double r3638059 = 2.0;
        double r3638060 = r3638059 / r3638056;
        double r3638061 = r3638058 - r3638060;
        double r3638062 = r3638056 - r3638055;
        double r3638063 = r3638055 / r3638062;
        double r3638064 = r3638061 + r3638063;
        return r3638064;
}
↓
double f(double x) {
        double r3638065 = 1.0;
        double r3638066 = x;
        double r3638067 = r3638066 - r3638065;
        double r3638068 = r3638065 / r3638067;
        double r3638069 = 2.0;
        double r3638070 = r3638066 + r3638065;
        double r3638071 = r3638070 * r3638066;
        double r3638072 = r3638069 / r3638071;
        double r3638073 = r3638068 * r3638072;
        return r3638073;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.4

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

  3. Applied frac-sub26.0

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

  4. Applied frac-add25.3

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

  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. Using strategy rm

  9. Applied div-inv0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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