Average Error: 9.6 → 0.3
Time: 22.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x - 1\right) \cdot x\right) \cdot \left(x + 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x - 1\right) \cdot x\right) \cdot \left(x + 1\right)}
double f(double x) {
        double r148260 = 1.0;
        double r148261 = x;
        double r148262 = r148261 + r148260;
        double r148263 = r148260 / r148262;
        double r148264 = 2.0;
        double r148265 = r148264 / r148261;
        double r148266 = r148263 - r148265;
        double r148267 = r148261 - r148260;
        double r148268 = r148260 / r148267;
        double r148269 = r148266 + r148268;
        return r148269;
}
↓
double f(double x) {
        double r148270 = 2.0;
        double r148271 = x;
        double r148272 = 1.0;
        double r148273 = r148271 - r148272;
        double r148274 = r148273 * r148271;
        double r148275 = r148271 + r148272;
        double r148276 = r148274 * r148275;
        double r148277 = r148270 / r148276;
        return r148277;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

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

  4. Applied frac-sub25.4

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

  5. Applied frac-add24.9

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

  6. Simplified24.9

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

  7. Simplified24.9

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

  8. Taylor expanded around 0 0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 
(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))))