Average Error: 20.4 → 0.4
Time: 2.4m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x} + \sqrt{x + 1}}\]
double f(double x) {
        double r12400235 = 1.0;
        double r12400236 = x;
        double r12400237 = sqrt(r12400236);
        double r12400238 = r12400235 / r12400237;
        double r12400239 = r12400236 + r12400235;
        double r12400240 = sqrt(r12400239);
        double r12400241 = r12400235 / r12400240;
        double r12400242 = r12400238 - r12400241;
        return r12400242;
}
↓
double f(double x) {
        double r12400243 = 1.0;
        double r12400244 = x;
        double r12400245 = r12400244 + r12400243;
        double r12400246 = sqrt(r12400245);
        double r12400247 = sqrt(r12400244);
        double r12400248 = r12400246 * r12400247;
        double r12400249 = r12400243 / r12400248;
        double r12400250 = r12400247 + r12400246;
        double r12400251 = r12400249 / r12400250;
        return r12400251;
}


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

Error

Bits error versus x

Target

Original20.4
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.4

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied frac-sub20.4

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

  4. Simplified20.4

    \[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  5. Using strategy rm

  6. Applied flip--20.2

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

  7. Applied associate-/l/20.2

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

  8. Simplified0.8

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

  10. Applied *-un-lft-identity0.8

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

  11. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
  12. Using strategy rm

  13. Applied un-div-inv0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))