Average Error: 10.1 → 0.1
Time: 15.4s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x + -1}}{\left(x + 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x + -1}}{\left(x + 1\right) \cdot x}
double f(double x) {
        double r5249991 = 1.0;
        double r5249992 = x;
        double r5249993 = r5249992 + r5249991;
        double r5249994 = r5249991 / r5249993;
        double r5249995 = 2.0;
        double r5249996 = r5249995 / r5249992;
        double r5249997 = r5249994 - r5249996;
        double r5249998 = r5249992 - r5249991;
        double r5249999 = r5249991 / r5249998;
        double r5250000 = r5249997 + r5249999;
        return r5250000;
}
↓
double f(double x) {
        double r5250001 = 2.0;
        double r5250002 = x;
        double r5250003 = -1.0;
        double r5250004 = r5250002 + r5250003;
        double r5250005 = r5250001 / r5250004;
        double r5250006 = 1.0;
        double r5250007 = r5250002 + r5250006;
        double r5250008 = r5250007 * r5250002;
        double r5250009 = r5250005 / r5250008;
        return r5250009;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.1

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

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

  5. Simplified25.1

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

  6. Simplified25.1

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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