Average Error: 9.4 → 0.1
Time: 34.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{\mathsf{fma}\left(x, x, x\right)} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{\mathsf{fma}\left(x, x, x\right)} \cdot \frac{2}{x - 1}
double f(double x) {
        double r6684486 = 1.0;
        double r6684487 = x;
        double r6684488 = r6684487 + r6684486;
        double r6684489 = r6684486 / r6684488;
        double r6684490 = 2.0;
        double r6684491 = r6684490 / r6684487;
        double r6684492 = r6684489 - r6684491;
        double r6684493 = r6684487 - r6684486;
        double r6684494 = r6684486 / r6684493;
        double r6684495 = r6684492 + r6684494;
        return r6684495;
}
↓
double f(double x) {
        double r6684496 = 1.0;
        double r6684497 = x;
        double r6684498 = fma(r6684497, r6684497, r6684497);
        double r6684499 = r6684496 / r6684498;
        double r6684500 = 2.0;
        double r6684501 = r6684497 - r6684496;
        double r6684502 = r6684500 / r6684501;
        double r6684503 = r6684499 * r6684502;
        return r6684503;
}

Error

Bits error versus x

Target

Original9.4
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.4

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

  3. Applied frac-sub26.0

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

  4. Applied frac-add25.3

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

  5. Simplified25.7

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

  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. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))