Average Error: 9.9 → 0.1
Time: 13.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}
double f(double x) {
        double r2226173 = 1.0;
        double r2226174 = x;
        double r2226175 = r2226174 + r2226173;
        double r2226176 = r2226173 / r2226175;
        double r2226177 = 2.0;
        double r2226178 = r2226177 / r2226174;
        double r2226179 = r2226176 - r2226178;
        double r2226180 = r2226174 - r2226173;
        double r2226181 = r2226173 / r2226180;
        double r2226182 = r2226179 + r2226181;
        return r2226182;
}
↓
double f(double x) {
        double r2226183 = 2.0;
        double r2226184 = x;
        double r2226185 = 1.0;
        double r2226186 = r2226184 + r2226185;
        double r2226187 = r2226183 / r2226186;
        double r2226188 = r2226187 / r2226184;
        double r2226189 = r2226184 - r2226185;
        double r2226190 = r2226188 / r2226189;
        return r2226190;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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-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-add25.5

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

  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. Using strategy rm

  7. Applied associate-/r*0.1

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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