Average Error: 9.8 → 0.1
Time: 1.2m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x - 1}}{\left(x + 1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x - 1}}{\left(x + 1\right) \cdot x}
double f(double x) {
        double r16487081 = 1.0;
        double r16487082 = x;
        double r16487083 = r16487082 + r16487081;
        double r16487084 = r16487081 / r16487083;
        double r16487085 = 2.0;
        double r16487086 = r16487085 / r16487082;
        double r16487087 = r16487084 - r16487086;
        double r16487088 = r16487082 - r16487081;
        double r16487089 = r16487081 / r16487088;
        double r16487090 = r16487087 + r16487089;
        return r16487090;
}
↓
double f(double x) {
        double r16487091 = 2.0;
        double r16487092 = x;
        double r16487093 = 1.0;
        double r16487094 = r16487092 - r16487093;
        double r16487095 = r16487091 / r16487094;
        double r16487096 = r16487092 + r16487093;
        double r16487097 = r16487096 * r16487092;
        double r16487098 = r16487095 / r16487097;
        return r16487098;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 +-commutative9.8

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

  5. Applied frac-sub25.8

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

  6. Applied frac-add25.0

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

  7. Simplified25.0

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

  8. Taylor expanded around inf 0.3

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

  10. Applied associate-/r*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019112 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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