Average Error: 10.2 → 0.3
Time: 20.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r4284063 = 1.0;
        double r4284064 = x;
        double r4284065 = r4284064 + r4284063;
        double r4284066 = r4284063 / r4284065;
        double r4284067 = 2.0;
        double r4284068 = r4284067 / r4284064;
        double r4284069 = r4284066 - r4284068;
        double r4284070 = r4284064 - r4284063;
        double r4284071 = r4284063 / r4284070;
        double r4284072 = r4284069 + r4284071;
        return r4284072;
}
↓
double f(double x) {
        double r4284073 = 2.0;
        double r4284074 = x;
        double r4284075 = 1.0;
        double r4284076 = r4284074 + r4284075;
        double r4284077 = r4284076 * r4284074;
        double r4284078 = r4284074 - r4284075;
        double r4284079 = r4284077 * r4284078;
        double r4284080 = r4284073 / r4284079;
        return r4284080;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 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-add25.3

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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