Average Error: 10.6 → 0.1
Time: 6.8s
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 r162562 = 1.0;
        double r162563 = x;
        double r162564 = r162563 + r162562;
        double r162565 = r162562 / r162564;
        double r162566 = 2.0;
        double r162567 = r162566 / r162563;
        double r162568 = r162565 - r162567;
        double r162569 = r162563 - r162562;
        double r162570 = r162562 / r162569;
        double r162571 = r162568 + r162570;
        return r162571;
}
↓
double f(double x) {
        double r162572 = 1.0;
        double r162573 = x;
        double r162574 = 1.0;
        double r162575 = r162573 + r162574;
        double r162576 = r162575 * r162573;
        double r162577 = r162572 / r162576;
        double r162578 = 2.0;
        double r162579 = r162573 - r162574;
        double r162580 = r162578 / r162579;
        double r162581 = r162577 * r162580;
        return r162581;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original10.6
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.6

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

  3. Applied frac-sub26.2

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

  4. Applied frac-add25.6

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

  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 2020045 
(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))))