Average Error: 9.7 → 0.1
Time: 18.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x - 1}}{1 + x}}{x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x - 1}}{1 + x}}{x}
double f(double x) {
        double r4416244 = 1.0;
        double r4416245 = x;
        double r4416246 = r4416245 + r4416244;
        double r4416247 = r4416244 / r4416246;
        double r4416248 = 2.0;
        double r4416249 = r4416248 / r4416245;
        double r4416250 = r4416247 - r4416249;
        double r4416251 = r4416245 - r4416244;
        double r4416252 = r4416244 / r4416251;
        double r4416253 = r4416250 + r4416252;
        return r4416253;
}
↓
double f(double x) {
        double r4416254 = 2.0;
        double r4416255 = x;
        double r4416256 = 1.0;
        double r4416257 = r4416255 - r4416256;
        double r4416258 = r4416254 / r4416257;
        double r4416259 = r4416256 + r4416255;
        double r4416260 = r4416258 / r4416259;
        double r4416261 = r4416260 / r4416255;
        return r4416261;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
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 frac-sub26.2

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

  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 0.2

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

  7. Applied *-un-lft-identity0.2

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

  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 *-un-lft-identity0.1

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

  11. Applied associate-*l*0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 
(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))))