Average Error: 9.7 → 0.1
Time: 13.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r2973041 = 1.0;
        double r2973042 = x;
        double r2973043 = r2973042 + r2973041;
        double r2973044 = r2973041 / r2973043;
        double r2973045 = 2.0;
        double r2973046 = r2973045 / r2973042;
        double r2973047 = r2973044 - r2973046;
        double r2973048 = r2973042 - r2973041;
        double r2973049 = r2973041 / r2973048;
        double r2973050 = r2973047 + r2973049;
        return r2973050;
}
↓
double f(double x) {
        double r2973051 = 2.0;
        double r2973052 = x;
        double r2973053 = 1.0;
        double r2973054 = r2973052 + r2973053;
        double r2973055 = r2973054 * r2973052;
        double r2973056 = r2973051 / r2973055;
        double r2973057 = r2973052 - r2973053;
        double r2973058 = r2973056 / r2973057;
        return r2973058;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied sub-neg9.7

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

  4. Simplified9.7

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

  6. Applied frac-add26.3

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

  7. Applied frac-add25.6

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

  8. Simplified25.6

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

  9. Taylor expanded around 0 0.3

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

  11. Applied associate-/r*0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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