Average Error: 10.0 → 0.1
Time: 4.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r119048 = 1.0;
        double r119049 = x;
        double r119050 = r119049 + r119048;
        double r119051 = r119048 / r119050;
        double r119052 = 2.0;
        double r119053 = r119052 / r119049;
        double r119054 = r119051 - r119053;
        double r119055 = r119049 - r119048;
        double r119056 = r119048 / r119055;
        double r119057 = r119054 + r119056;
        return r119057;
}
↓
double f(double x) {
        double r119058 = 2.0;
        double r119059 = x;
        double r119060 = 1.0;
        double r119061 = r119059 + r119060;
        double r119062 = r119061 * r119059;
        double r119063 = r119058 / r119062;
        double r119064 = r119059 - r119060;
        double r119065 = r119063 / r119064;
        return r119065;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-add25.7

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

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019344 
(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))))