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 r210295 = 1.0;
        double r210296 = x;
        double r210297 = sqrt(r210296);
        double r210298 = r210295 / r210297;
        double r210299 = r210296 + r210295;
        double r210300 = sqrt(r210299);
        double r210301 = r210295 / r210300;
        double r210302 = r210298 - r210301;
        return r210302;
}
↓
double f(double x) {
        double r210303 = 1.0;
        double r210304 = x;
        double r210305 = sqrt(r210304);
        double r210306 = r210303 / r210305;
        double r210307 = r210304 + r210303;
        double r210308 = sqrt(r210307);
        double r210309 = r210305 * r210308;
        double r210310 = r210309 + r210307;
        double r210311 = r210303 / r210310;
        double r210312 = r210306 * r210311;
        return r210312;
}

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)))))