Average Error: 10.2 → 0.1
Time: 4.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r140258 = 1.0;
        double r140259 = x;
        double r140260 = r140259 + r140258;
        double r140261 = r140258 / r140260;
        double r140262 = 2.0;
        double r140263 = r140262 / r140259;
        double r140264 = r140261 - r140263;
        double r140265 = r140259 - r140258;
        double r140266 = r140258 / r140265;
        double r140267 = r140264 + r140266;
        return r140267;
}
↓
double f(double x) {
        double r140268 = 2.0;
        double r140269 = x;
        double r140270 = 1.0;
        double r140271 = r140269 + r140270;
        double r140272 = r140271 * r140269;
        double r140273 = r140268 / r140272;
        double r140274 = r140269 - r140270;
        double r140275 = r140273 / r140274;
        return r140275;
}

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 *-un-lft-identity10.2

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

  4. Applied add-sqr-sqrt10.2

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

  5. Applied times-frac10.2

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

  6. Simplified10.2

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

  8. Applied associate-*r/10.2

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

  9. Applied frac-sub26.6

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

  10. Applied frac-add26.1

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

  11. Simplified26.1

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

  12. Taylor expanded around 0 0.3

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

  14. Applied associate-/r*0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 
(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))))