Average Error: 9.8 → 0.1
Time: 32.4s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[2 \cdot \frac{\frac{1}{x}}{x \cdot x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{\frac{1}{x}}{x \cdot x - 1}
double f(double x) {
        double r3828752 = 1.0;
        double r3828753 = x;
        double r3828754 = r3828753 + r3828752;
        double r3828755 = r3828752 / r3828754;
        double r3828756 = 2.0;
        double r3828757 = r3828756 / r3828753;
        double r3828758 = r3828755 - r3828757;
        double r3828759 = r3828753 - r3828752;
        double r3828760 = r3828752 / r3828759;
        double r3828761 = r3828758 + r3828760;
        return r3828761;
}
↓
double f(double x) {
        double r3828762 = 2.0;
        double r3828763 = 1.0;
        double r3828764 = x;
        double r3828765 = r3828763 / r3828764;
        double r3828766 = r3828764 * r3828764;
        double r3828767 = r3828766 - r3828763;
        double r3828768 = r3828765 / r3828767;
        double r3828769 = r3828762 * r3828768;
        return r3828769;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.8

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub26.0

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

  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 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 div-inv0.3

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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