Average Error: 9.7 → 0.1
Time: 15.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{x - 1} \cdot \frac{1}{\left(x + 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{x - 1} \cdot \frac{1}{\left(x + 1\right) \cdot x}
double f(double x) {
        double r7647671 = 1.0;
        double r7647672 = x;
        double r7647673 = r7647672 + r7647671;
        double r7647674 = r7647671 / r7647673;
        double r7647675 = 2.0;
        double r7647676 = r7647675 / r7647672;
        double r7647677 = r7647674 - r7647676;
        double r7647678 = r7647672 - r7647671;
        double r7647679 = r7647671 / r7647678;
        double r7647680 = r7647677 + r7647679;
        return r7647680;
}
↓
double f(double x) {
        double r7647681 = 2.0;
        double r7647682 = x;
        double r7647683 = 1.0;
        double r7647684 = r7647682 - r7647683;
        double r7647685 = r7647681 / r7647684;
        double r7647686 = 1.0;
        double r7647687 = r7647682 + r7647683;
        double r7647688 = r7647687 * r7647682;
        double r7647689 = r7647686 / r7647688;
        double r7647690 = r7647685 * r7647689;
        return r7647690;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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 *-un-lft-identity0.2

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

  8. Applied times-frac0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 
(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))))