Average Error: 9.6 → 0.3
Time: 18.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{x \cdot \left(x \cdot x - 1\right)}
double f(double x) {
        double r117314 = 1.0;
        double r117315 = x;
        double r117316 = r117315 + r117314;
        double r117317 = r117314 / r117316;
        double r117318 = 2.0;
        double r117319 = r117318 / r117315;
        double r117320 = r117317 - r117319;
        double r117321 = r117315 - r117314;
        double r117322 = r117314 / r117321;
        double r117323 = r117320 + r117322;
        return r117323;
}
↓
double f(double x) {
        double r117324 = 2.0;
        double r117325 = x;
        double r117326 = r117325 * r117325;
        double r117327 = 1.0;
        double r117328 = r117326 - r117327;
        double r117329 = r117325 * r117328;
        double r117330 = r117324 / r117329;
        return r117330;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.6

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

  4. Applied frac-sub25.4

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

  5. Applied frac-add24.9

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

  6. Simplified24.9

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

  7. Simplified24.9

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

  8. Taylor expanded around 0 0.3

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

  9. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - 1 \cdot x}}\]

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))