Average Error: 9.7 → 0.1
Time: 4.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\frac{x + 1}{1} \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\frac{x + 1}{1} \cdot x}}{x - 1}
double f(double x) {
        double r225123 = 1.0;
        double r225124 = x;
        double r225125 = r225124 + r225123;
        double r225126 = r225123 / r225125;
        double r225127 = 2.0;
        double r225128 = r225127 / r225124;
        double r225129 = r225126 - r225128;
        double r225130 = r225124 - r225123;
        double r225131 = r225123 / r225130;
        double r225132 = r225129 + r225131;
        return r225132;
}
↓
double f(double x) {
        double r225133 = 2.0;
        double r225134 = x;
        double r225135 = 1.0;
        double r225136 = r225134 + r225135;
        double r225137 = r225136 / r225135;
        double r225138 = r225137 * r225134;
        double r225139 = r225133 / r225138;
        double r225140 = r225134 - r225135;
        double r225141 = r225139 / r225140;
        return r225141;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 clear-num9.7

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

  5. Applied frac-sub26.0

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

  6. Applied frac-add25.5

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

  7. Taylor expanded around 0 0.2

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 
(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))))