Average Error: 20.0 → 0.3
Time: 9.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1} + \left(x + 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1} + \left(x + 1\right)}
double f(double x) {
        double r158340 = 1.0;
        double r158341 = x;
        double r158342 = sqrt(r158341);
        double r158343 = r158340 / r158342;
        double r158344 = r158341 + r158340;
        double r158345 = sqrt(r158344);
        double r158346 = r158340 / r158345;
        double r158347 = r158343 - r158346;
        return r158347;
}
↓
double f(double x) {
        double r158348 = 1.0;
        double r158349 = x;
        double r158350 = sqrt(r158349);
        double r158351 = r158348 / r158350;
        double r158352 = r158349 + r158348;
        double r158353 = sqrt(r158352);
        double r158354 = r158350 * r158353;
        double r158355 = r158354 + r158352;
        double r158356 = r158348 / r158355;
        double r158357 = r158351 * r158356;
        return r158357;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.8

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

  7. Simplified19.2

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

  8. Taylor expanded around 0 0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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