Average Error: 9.9 → 0.1
Time: 13.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{1 + x}}{x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{1 + x}}{x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r82122 = 1.0;
        double r82123 = x;
        double r82124 = r82123 + r82122;
        double r82125 = r82122 / r82124;
        double r82126 = 2.0;
        double r82127 = r82126 / r82123;
        double r82128 = r82125 - r82127;
        double r82129 = r82123 - r82122;
        double r82130 = r82122 / r82129;
        double r82131 = r82128 + r82130;
        return r82131;
}
↓
double f(double x) {
        double r82132 = 1.0;
        double r82133 = 1.0;
        double r82134 = x;
        double r82135 = r82133 + r82134;
        double r82136 = r82132 / r82135;
        double r82137 = r82136 / r82134;
        double r82138 = 2.0;
        double r82139 = r82134 - r82133;
        double r82140 = r82138 / r82139;
        double r82141 = r82137 * r82140;
        return r82141;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-add25.8

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

  5. 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 *-un-lft-identity0.3

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

  8. Applied times-frac0.1

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

  10. Applied associate-/r*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 
(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))))