Average Error: 9.9 → 0.3
Time: 19.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r2811050 = 1.0;
        double r2811051 = x;
        double r2811052 = r2811051 + r2811050;
        double r2811053 = r2811050 / r2811052;
        double r2811054 = 2.0;
        double r2811055 = r2811054 / r2811051;
        double r2811056 = r2811053 - r2811055;
        double r2811057 = r2811051 - r2811050;
        double r2811058 = r2811050 / r2811057;
        double r2811059 = r2811056 + r2811058;
        return r2811059;
}
↓
double f(double x) {
        double r2811060 = 2.0;
        double r2811061 = x;
        double r2811062 = 1.0;
        double r2811063 = r2811061 + r2811062;
        double r2811064 = r2811063 * r2811061;
        double r2811065 = r2811061 - r2811062;
        double r2811066 = r2811064 * r2811065;
        double r2811067 = r2811060 / r2811066;
        return r2811067;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

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-sub25.9

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

  4. Applied frac-add25.1

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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