Average Error: 9.9 → 0.1
Time: 19.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r100500 = 1.0;
        double r100501 = x;
        double r100502 = r100501 + r100500;
        double r100503 = r100500 / r100502;
        double r100504 = 2.0;
        double r100505 = r100504 / r100501;
        double r100506 = r100503 - r100505;
        double r100507 = r100501 - r100500;
        double r100508 = r100500 / r100507;
        double r100509 = r100506 + r100508;
        return r100509;
}
↓
double f(double x) {
        double r100510 = 1.0;
        double r100511 = x;
        double r100512 = 1.0;
        double r100513 = r100511 + r100512;
        double r100514 = r100510 / r100513;
        double r100515 = r100514 / r100511;
        double r100516 = 2.0;
        double r100517 = r100511 - r100512;
        double r100518 = r100516 / r100517;
        double r100519 = r100515 * r100518;
        return r100519;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-add25.8

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

  5. Simplified26.1

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

  6. Taylor expanded around 0 0.3

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

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

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

  9. Applied times-frac0.1

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

  11. Applied associate-/r*0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))