Average Error: 9.8 → 0.2
Time: 8.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - 1 \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - 1 \cdot x}
double f(double x) {
        double r112009 = 1.0;
        double r112010 = x;
        double r112011 = r112010 + r112009;
        double r112012 = r112009 / r112011;
        double r112013 = 2.0;
        double r112014 = r112013 / r112010;
        double r112015 = r112012 - r112014;
        double r112016 = r112010 - r112009;
        double r112017 = r112009 / r112016;
        double r112018 = r112015 + r112017;
        return r112018;
}
↓
double f(double x) {
        double r112019 = 2.0;
        double r112020 = x;
        double r112021 = 3.0;
        double r112022 = pow(r112020, r112021);
        double r112023 = 1.0;
        double r112024 = r112023 * r112020;
        double r112025 = r112022 - r112024;
        double r112026 = r112019 / r112025;
        return r112026;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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-sub25.9

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

  4. Applied frac-add25.3

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

  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. Taylor expanded around 0 0.2

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - 1 \cdot x}}\]

  7. Final simplification0.2

    \[\leadsto \frac{2}{{x}^{3} - 1 \cdot x}\]

Reproduce

herbie shell --seed 2020042 
(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))))