Average Error: 10.2 → 0.3
Time: 24.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}
double f(double x) {
        double r3851997 = 1.0;
        double r3851998 = x;
        double r3851999 = r3851998 + r3851997;
        double r3852000 = r3851997 / r3851999;
        double r3852001 = 2.0;
        double r3852002 = r3852001 / r3851998;
        double r3852003 = r3852000 - r3852002;
        double r3852004 = r3851998 - r3851997;
        double r3852005 = r3851997 / r3852004;
        double r3852006 = r3852003 + r3852005;
        return r3852006;
}
↓
double f(double x) {
        double r3852007 = 2.0;
        double r3852008 = x;
        double r3852009 = 1.0;
        double r3852010 = r3852008 + r3852009;
        double r3852011 = r3852008 - r3852009;
        double r3852012 = r3852008 * r3852011;
        double r3852013 = r3852010 * r3852012;
        double r3852014 = r3852007 / r3852013;
        return r3852014;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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-sub26.1

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

  4. Applied frac-add25.6

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

  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. Using strategy rm

  7. Applied associate-/r*0.1

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

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

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

  10. Applied flip-+0.1

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

  11. Applied associate-*l/0.3

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

  12. Applied associate-/r/0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))))