Average Error: 9.7 → 0.1
Time: 4.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r132467 = 1.0;
        double r132468 = x;
        double r132469 = r132468 + r132467;
        double r132470 = r132467 / r132469;
        double r132471 = 2.0;
        double r132472 = r132471 / r132468;
        double r132473 = r132470 - r132472;
        double r132474 = r132468 - r132467;
        double r132475 = r132467 / r132474;
        double r132476 = r132473 + r132475;
        return r132476;
}
↓
double f(double x) {
        double r132477 = 1.0;
        double r132478 = x;
        double r132479 = 1.0;
        double r132480 = r132478 + r132479;
        double r132481 = r132480 * r132478;
        double r132482 = r132477 / r132481;
        double r132483 = 2.0;
        double r132484 = r132478 - r132479;
        double r132485 = r132483 / r132484;
        double r132486 = r132482 * r132485;
        return r132486;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original9.7
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub26.1

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

  4. Applied frac-add25.7

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

  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. Using strategy rm

  7. Applied *-un-lft-identity0.3

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

  8. Applied times-frac0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 
(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))))