Average Error: 9.9 → 0.1
Time: 16.4s
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 r5321724 = 1.0;
        double r5321725 = x;
        double r5321726 = r5321725 + r5321724;
        double r5321727 = r5321724 / r5321726;
        double r5321728 = 2.0;
        double r5321729 = r5321728 / r5321725;
        double r5321730 = r5321727 - r5321729;
        double r5321731 = r5321725 - r5321724;
        double r5321732 = r5321724 / r5321731;
        double r5321733 = r5321730 + r5321732;
        return r5321733;
}
↓
double f(double x) {
        double r5321734 = 2.0;
        double r5321735 = x;
        double r5321736 = 1.0;
        double r5321737 = r5321735 - r5321736;
        double r5321738 = r5321734 / r5321737;
        double r5321739 = r5321735 + r5321736;
        double r5321740 = r5321739 * r5321735;
        double r5321741 = r5321738 / r5321740;
        return r5321741;
}

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-sub26.3

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

  4. Applied frac-add25.4

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

  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 add-sqr-sqrt1.0

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

  8. Applied times-frac0.5

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

  10. Applied associate-*l/0.7

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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