Average Error: 9.9 → 0.1
Time: 2.9m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{(x \cdot x + -1)_*}}{x}\]
double f(double x) {
        double r12123371 = 1.0;
        double r12123372 = x;
        double r12123373 = r12123372 + r12123371;
        double r12123374 = r12123371 / r12123373;
        double r12123375 = 2.0;
        double r12123376 = r12123375 / r12123372;
        double r12123377 = r12123374 - r12123376;
        double r12123378 = r12123372 - r12123371;
        double r12123379 = r12123371 / r12123378;
        double r12123380 = r12123377 + r12123379;
        return r12123380;
}
↓
double f(double x) {
        double r12123381 = 2.0;
        double r12123382 = x;
        double r12123383 = -1.0;
        double r12123384 = fma(r12123382, r12123382, r12123383);
        double r12123385 = r12123381 / r12123384;
        double r12123386 = r12123385 / r12123382;
        return r12123386;
}


\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{(x \cdot x + -1)_*}}{x}

Error

Bits error versus x

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-sub26.0

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

  4. Applied frac-add25.4

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

  5. Simplified25.7

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

  6. Simplified25.7

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

    \[\leadsto \frac{\frac{2}{(x \cdot x + -1)_*}}{x}\]

Reproduce

herbie shell --seed 2019102 +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))))