Average Error: 30.2 → 0.3
Time: 8.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \left|\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right|\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \left|\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right|
double f(double x) {
        double r121260 = x;
        double r121261 = 1.0;
        double r121262 = r121260 + r121261;
        double r121263 = sqrt(r121262);
        double r121264 = sqrt(r121260);
        double r121265 = r121263 - r121264;
        return r121265;
}
↓
double f(double x) {
        double r121266 = 1.0;
        double r121267 = x;
        double r121268 = r121267 + r121266;
        double r121269 = sqrt(r121268);
        double r121270 = sqrt(r121267);
        double r121271 = r121269 + r121270;
        double r121272 = r121266 / r121271;
        double r121273 = sqrt(r121272);
        double r121274 = sqrt(r121266);
        double r121275 = sqrt(r121271);
        double r121276 = r121274 / r121275;
        double r121277 = fabs(r121276);
        double r121278 = r121273 * r121277;
        return r121278;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.0

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied add-sqr-sqrt0.3

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

  10. Applied times-frac0.3

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

  11. Applied rem-sqrt-square0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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