Average Error: 9.9 → 0.1
Time: 12.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}
double f(double x) {
        double r79880 = 1.0;
        double r79881 = x;
        double r79882 = r79881 + r79880;
        double r79883 = r79880 / r79882;
        double r79884 = 2.0;
        double r79885 = r79884 / r79881;
        double r79886 = r79883 - r79885;
        double r79887 = r79881 - r79880;
        double r79888 = r79880 / r79887;
        double r79889 = r79886 + r79888;
        return r79889;
}
↓
double f(double x) {
        double r79890 = 2.0;
        double r79891 = x;
        double r79892 = 1.0;
        double r79893 = r79891 + r79892;
        double r79894 = r79890 / r79893;
        double r79895 = r79894 / r79891;
        double r79896 = r79891 - r79892;
        double r79897 = r79895 / r79896;
        return r79897;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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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 associate-/r*0.1

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

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))))