Average Error: 10.2 → 0.1
Time: 14.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x + -1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x + -1}
double f(double x) {
        double r1133751 = 1.0;
        double r1133752 = x;
        double r1133753 = r1133752 + r1133751;
        double r1133754 = r1133751 / r1133753;
        double r1133755 = 2.0;
        double r1133756 = r1133755 / r1133752;
        double r1133757 = r1133754 - r1133756;
        double r1133758 = r1133752 - r1133751;
        double r1133759 = r1133751 / r1133758;
        double r1133760 = r1133757 + r1133759;
        return r1133760;
}
↓
double f(double x) {
        double r1133761 = 1.0;
        double r1133762 = x;
        double r1133763 = r1133762 + r1133761;
        double r1133764 = r1133761 / r1133763;
        double r1133765 = r1133764 / r1133762;
        double r1133766 = 2.0;
        double r1133767 = -1.0;
        double r1133768 = r1133762 + r1133767;
        double r1133769 = r1133766 / r1133768;
        double r1133770 = r1133765 * r1133769;
        return r1133770;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.2

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

  3. Applied frac-sub26.4

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

  4. Applied frac-add25.8

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

  5. Simplified25.8

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

  6. Simplified25.8

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

  7. Taylor expanded around inf 0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied times-frac0.1

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

  12. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x}} \cdot \frac{2}{x + -1}\]

  13. Final simplification0.1

    \[\leadsto \frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x + -1}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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