Average Error: 10.0 → 0.1
Time: 1.4m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[2 \cdot \frac{\frac{1}{x}}{-1 + x \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{\frac{1}{x}}{-1 + x \cdot x}
double f(double x) {
        double r10913014 = 1.0;
        double r10913015 = x;
        double r10913016 = r10913015 + r10913014;
        double r10913017 = r10913014 / r10913016;
        double r10913018 = 2.0;
        double r10913019 = r10913018 / r10913015;
        double r10913020 = r10913017 - r10913019;
        double r10913021 = r10913015 - r10913014;
        double r10913022 = r10913014 / r10913021;
        double r10913023 = r10913020 + r10913022;
        return r10913023;
}
↓
double f(double x) {
        double r10913024 = 2.0;
        double r10913025 = 1.0;
        double r10913026 = x;
        double r10913027 = r10913025 / r10913026;
        double r10913028 = -1.0;
        double r10913029 = r10913026 * r10913026;
        double r10913030 = r10913028 + r10913029;
        double r10913031 = r10913027 / r10913030;
        double r10913032 = r10913024 * r10913031;
        return r10913032;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

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

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

  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 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 associate-/l*0.8

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

  10. Applied div-inv0.8

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

  11. Applied *-un-lft-identity0.8

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

  12. Applied sqrt-prod0.8

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

  13. Applied times-frac1.0

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

  14. Simplified0.8

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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