Average Error: 10.2 → 0.1
Time: 11.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x + -1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x + -1}
double f(double x) {
        double r2008241 = 1.0;
        double r2008242 = x;
        double r2008243 = r2008242 + r2008241;
        double r2008244 = r2008241 / r2008243;
        double r2008245 = 2.0;
        double r2008246 = r2008245 / r2008242;
        double r2008247 = r2008244 - r2008246;
        double r2008248 = r2008242 - r2008241;
        double r2008249 = r2008241 / r2008248;
        double r2008250 = r2008247 + r2008249;
        return r2008250;
}
↓
double f(double x) {
        double r2008251 = 1.0;
        double r2008252 = x;
        double r2008253 = r2008252 + r2008251;
        double r2008254 = r2008251 / r2008253;
        double r2008255 = r2008254 / r2008252;
        double r2008256 = 2.0;
        double r2008257 = -1.0;
        double r2008258 = r2008252 + r2008257;
        double r2008259 = r2008256 / r2008258;
        double r2008260 = r2008255 * r2008259;
        return r2008260;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.2

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

  3. Applied frac-sub26.4

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

  4. Applied frac-add25.8

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

  5. Simplified25.8

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

  6. Simplified25.8

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

  7. Taylor expanded around 0 0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied times-frac0.1

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

  12. Applied associate-/r*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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