Average Error: 0.6 → 0.6
Time: 18.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
double f(double x) {
        double r3114581 = 1.0;
        double r3114582 = x;
        double r3114583 = sqrt(r3114582);
        double r3114584 = r3114581 / r3114583;
        double r3114585 = r3114582 + r3114581;
        double r3114586 = sqrt(r3114585);
        double r3114587 = r3114581 / r3114586;
        double r3114588 = r3114584 - r3114587;
        return r3114588;
}
↓
double f(double x) {
        double r3114589 = 1.0;
        double r3114590 = x;
        double r3114591 = sqrt(r3114590);
        double r3114592 = r3114589 / r3114591;
        double r3114593 = r3114592 * r3114592;
        double r3114594 = r3114589 * r3114589;
        double r3114595 = r3114590 + r3114589;
        double r3114596 = sqrt(r3114595);
        double r3114597 = r3114596 * r3114596;
        double r3114598 = r3114594 / r3114597;
        double r3114599 = r3114593 - r3114598;
        double r3114600 = r3114589 / r3114596;
        double r3114601 = r3114592 + r3114600;
        double r3114602 = r3114599 / r3114601;
        return r3114602;
}


\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}

Error

Bits error versus x

Derivation

  1. Initial program 0.6

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

  3. Applied p16-flip--0.7

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

  5. Applied associate-*r/0.7

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

  7. Applied associate-*l/0.7

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

  8. Applied associate-/l/0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  (-.p16 (/.p16 (real->posit16 1) (sqrt.p16 x)) (/.p16 (real->posit16 1) (sqrt.p16 (+.p16 x (real->posit16 1))))))