Average Error: 9.8 → 0.2
Time: 8.7s
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 r108219 = 1.0;
        double r108220 = x;
        double r108221 = r108220 + r108219;
        double r108222 = r108219 / r108221;
        double r108223 = 2.0;
        double r108224 = r108223 / r108220;
        double r108225 = r108222 - r108224;
        double r108226 = r108220 - r108219;
        double r108227 = r108219 / r108226;
        double r108228 = r108225 + r108227;
        return r108228;
}
↓
double f(double x) {
        double r108229 = 2.0;
        double r108230 = x;
        double r108231 = 3.0;
        double r108232 = pow(r108230, r108231);
        double r108233 = 1.0;
        double r108234 = r108233 * r108230;
        double r108235 = r108232 - r108234;
        double r108236 = r108229 / r108235;
        return r108236;
}

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))))