Average Error: 10.2 → 0.1
Time: 20.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x \cdot \left(1 + x\right)}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x \cdot \left(1 + x\right)}}{x - 1}
double f(double x) {
        double r1651130 = 1.0;
        double r1651131 = x;
        double r1651132 = r1651131 + r1651130;
        double r1651133 = r1651130 / r1651132;
        double r1651134 = 2.0;
        double r1651135 = r1651134 / r1651131;
        double r1651136 = r1651133 - r1651135;
        double r1651137 = r1651131 - r1651130;
        double r1651138 = r1651130 / r1651137;
        double r1651139 = r1651136 + r1651138;
        return r1651139;
}
↓
double f(double x) {
        double r1651140 = 2.0;
        double r1651141 = x;
        double r1651142 = 1.0;
        double r1651143 = r1651142 + r1651141;
        double r1651144 = r1651141 * r1651143;
        double r1651145 = r1651140 / r1651144;
        double r1651146 = r1651141 - r1651142;
        double r1651147 = r1651145 / r1651146;
        return r1651147;
}

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. Taylor expanded around inf 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

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

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

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

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