Average Error: 9.6 → 0.3
Time: 3.4s
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 r129017 = 1.0;
        double r129018 = x;
        double r129019 = r129018 + r129017;
        double r129020 = r129017 / r129019;
        double r129021 = 2.0;
        double r129022 = r129021 / r129018;
        double r129023 = r129020 - r129022;
        double r129024 = r129018 - r129017;
        double r129025 = r129017 / r129024;
        double r129026 = r129023 + r129025;
        return r129026;
}
↓
double f(double x) {
        double r129027 = 2.0;
        double r129028 = x;
        double r129029 = 1.0;
        double r129030 = r129028 + r129029;
        double r129031 = r129030 * r129028;
        double r129032 = r129028 - r129029;
        double r129033 = r129031 * r129032;
        double r129034 = r129027 / r129033;
        return r129034;
}

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. Using strategy rm

  3. Applied frac-sub25.9

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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