Average Error: 10.1 → 0.1
Time: 45.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1}
double f(double x) {
        double r3382554 = 1.0;
        double r3382555 = x;
        double r3382556 = r3382555 + r3382554;
        double r3382557 = r3382554 / r3382556;
        double r3382558 = 2.0;
        double r3382559 = r3382558 / r3382555;
        double r3382560 = r3382557 - r3382559;
        double r3382561 = r3382555 - r3382554;
        double r3382562 = r3382554 / r3382561;
        double r3382563 = r3382560 + r3382562;
        return r3382563;
}
↓
double f(double x) {
        double r3382564 = 2.0;
        double r3382565 = x;
        double r3382566 = fma(r3382565, r3382565, r3382565);
        double r3382567 = r3382564 / r3382566;
        double r3382568 = 1.0;
        double r3382569 = r3382565 - r3382568;
        double r3382570 = r3382567 / r3382569;
        return r3382570;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 10.1

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

  3. Applied frac-sub26.5

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

  5. Applied frac-add25.8

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

  6. Taylor expanded around inf 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 associate-/r*0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019143 +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))))