Average Error: 10.1 → 0.3
Time: 18.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot x\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot x\right)}
double f(double x) {
        double r4317330 = 1.0;
        double r4317331 = x;
        double r4317332 = r4317331 + r4317330;
        double r4317333 = r4317330 / r4317332;
        double r4317334 = 2.0;
        double r4317335 = r4317334 / r4317331;
        double r4317336 = r4317333 - r4317335;
        double r4317337 = r4317331 - r4317330;
        double r4317338 = r4317330 / r4317337;
        double r4317339 = r4317336 + r4317338;
        return r4317339;
}
↓
double f(double x) {
        double r4317340 = 2.0;
        double r4317341 = x;
        double r4317342 = -1.0;
        double r4317343 = r4317341 + r4317342;
        double r4317344 = 1.0;
        double r4317345 = r4317341 + r4317344;
        double r4317346 = r4317345 * r4317341;
        double r4317347 = r4317343 * r4317346;
        double r4317348 = r4317340 / r4317347;
        return r4317348;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.1

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

  3. Applied frac-sub25.5

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

  4. Applied frac-add24.9

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

  5. Simplified25.3

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

  6. Simplified25.3

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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