Average Error: 0.6 → 0.6
Time: 17.7s
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 r3898227 = 1.0;
        double r3898228 = x;
        double r3898229 = sqrt(r3898228);
        double r3898230 = r3898227 / r3898229;
        double r3898231 = r3898228 + r3898227;
        double r3898232 = sqrt(r3898231);
        double r3898233 = r3898227 / r3898232;
        double r3898234 = r3898230 - r3898233;
        return r3898234;
}
↓
double f(double x) {
        double r3898235 = 1.0;
        double r3898236 = x;
        double r3898237 = sqrt(r3898236);
        double r3898238 = r3898235 / r3898237;
        double r3898239 = r3898238 * r3898238;
        double r3898240 = r3898235 * r3898235;
        double r3898241 = r3898236 + r3898235;
        double r3898242 = sqrt(r3898241);
        double r3898243 = r3898242 * r3898242;
        double r3898244 = r3898240 / r3898243;
        double r3898245 = r3898239 - r3898244;
        double r3898246 = r3898235 / r3898242;
        double r3898247 = r3898238 + r3898246;
        double r3898248 = r3898245 / r3898247;
        return r3898248;
}


\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 +o rules:numerics
(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))))))