Average Error: 10.0 → 0.1
Time: 16.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{1 + x}}{x - 1}}{x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{1 + x}}{x - 1}}{x}
double f(double x) {
        double r81194 = 1.0;
        double r81195 = x;
        double r81196 = r81195 + r81194;
        double r81197 = r81194 / r81196;
        double r81198 = 2.0;
        double r81199 = r81198 / r81195;
        double r81200 = r81197 - r81199;
        double r81201 = r81195 - r81194;
        double r81202 = r81194 / r81201;
        double r81203 = r81200 + r81202;
        return r81203;
}
↓
double f(double x) {
        double r81204 = 2.0;
        double r81205 = 1.0;
        double r81206 = x;
        double r81207 = r81205 + r81206;
        double r81208 = r81204 / r81207;
        double r81209 = r81206 - r81205;
        double r81210 = r81208 / r81209;
        double r81211 = r81210 / r81206;
        return r81211;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-add26.1

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

  5. Simplified26.1

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

  6. Simplified26.1

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{1 + x}}{x - 1}}}{x}\]

  11. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{2}{1 + x}}{x - 1}}{x}\]

Reproduce

herbie shell --seed 2019196 
(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))))