Average Error: 10.2 → 0.3
Time: 16.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - x \cdot 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - x \cdot 1}
double f(double x) {
        double r134035 = 1.0;
        double r134036 = x;
        double r134037 = r134036 + r134035;
        double r134038 = r134035 / r134037;
        double r134039 = 2.0;
        double r134040 = r134039 / r134036;
        double r134041 = r134038 - r134040;
        double r134042 = r134036 - r134035;
        double r134043 = r134035 / r134042;
        double r134044 = r134041 + r134043;
        return r134044;
}
↓
double f(double x) {
        double r134045 = 2.0;
        double r134046 = x;
        double r134047 = 3.0;
        double r134048 = pow(r134046, r134047);
        double r134049 = 1.0;
        double r134050 = r134046 * r134049;
        double r134051 = r134048 - r134050;
        double r134052 = r134045 / r134051;
        return r134052;
}

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. Simplified10.2

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

  4. Applied frac-sub26.1

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

  5. Applied frac-add25.6

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

  6. Simplified25.6

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

  7. Taylor expanded around 0 0.3

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

  8. Taylor expanded around 0 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))