Average Error: 9.8 → 0.1
Time: 3.6s
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 r109438 = 1.0;
        double r109439 = x;
        double r109440 = r109439 + r109438;
        double r109441 = r109438 / r109440;
        double r109442 = 2.0;
        double r109443 = r109442 / r109439;
        double r109444 = r109441 - r109443;
        double r109445 = r109439 - r109438;
        double r109446 = r109438 / r109445;
        double r109447 = r109444 + r109446;
        return r109447;
}
↓
double f(double x) {
        double r109448 = 1.0;
        double r109449 = x;
        double r109450 = 1.0;
        double r109451 = r109449 + r109450;
        double r109452 = r109451 * r109449;
        double r109453 = r109448 / r109452;
        double r109454 = 2.0;
        double r109455 = r109449 - r109450;
        double r109456 = r109454 / r109455;
        double r109457 = r109453 * r109456;
        return r109457;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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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 frac-sub25.8

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

  4. Applied frac-add25.2

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

  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 2019362 
(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))))