Average Error: 9.6 → 0.1
Time: 17.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1} \cdot \frac{2}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1} \cdot \frac{2}{x}}{x - 1}
double f(double x) {
        double r166750 = 1.0;
        double r166751 = x;
        double r166752 = r166751 + r166750;
        double r166753 = r166750 / r166752;
        double r166754 = 2.0;
        double r166755 = r166754 / r166751;
        double r166756 = r166753 - r166755;
        double r166757 = r166751 - r166750;
        double r166758 = r166750 / r166757;
        double r166759 = r166756 + r166758;
        return r166759;
}
↓
double f(double x) {
        double r166760 = 1.0;
        double r166761 = x;
        double r166762 = 1.0;
        double r166763 = r166761 + r166762;
        double r166764 = r166760 / r166763;
        double r166765 = 2.0;
        double r166766 = r166765 / r166761;
        double r166767 = r166764 * r166766;
        double r166768 = r166761 - r166762;
        double r166769 = r166767 / r166768;
        return r166769;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

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-sub25.7

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

  4. Applied frac-add25.0

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

  5. Simplified25.5

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

  6. Taylor expanded around 0 0.3

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

  8. Applied associate-/r*0.1

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

  10. Applied *-un-lft-identity0.1

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

  11. Applied times-frac0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))))