Average Error: 20.4 → 0.4
Time: 34.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}\]
double f(double x) {
        double r13647236 = 1.0;
        double r13647237 = x;
        double r13647238 = sqrt(r13647237);
        double r13647239 = r13647236 / r13647238;
        double r13647240 = r13647237 + r13647236;
        double r13647241 = sqrt(r13647240);
        double r13647242 = r13647236 / r13647241;
        double r13647243 = r13647239 - r13647242;
        return r13647243;
}
↓
double f(double x) {
        double r13647244 = 1.0;
        double r13647245 = x;
        double r13647246 = sqrt(r13647245);
        double r13647247 = r13647244 / r13647246;
        double r13647248 = r13647245 + r13647244;
        double r13647249 = sqrt(r13647248);
        double r13647250 = r13647249 + r13647246;
        double r13647251 = r13647244 / r13647249;
        double r13647252 = r13647250 / r13647251;
        double r13647253 = r13647247 / r13647252;
        return r13647253;
}


\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\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 associate-/r*0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied times-frac0.4

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

  14. Applied associate-/l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 
(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)))))