Average Error: 19.6 → 0.3
Time: 4.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{\sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{\sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}
double f(double x) {
        double r136278 = 1.0;
        double r136279 = x;
        double r136280 = sqrt(r136279);
        double r136281 = r136278 / r136280;
        double r136282 = r136279 + r136278;
        double r136283 = sqrt(r136282);
        double r136284 = r136278 / r136283;
        double r136285 = r136281 - r136284;
        return r136285;
}
↓
double f(double x) {
        double r136286 = 1.0;
        double r136287 = r136286 * r136286;
        double r136288 = x;
        double r136289 = sqrt(r136288);
        double r136290 = r136287 / r136289;
        double r136291 = r136288 + r136286;
        double r136292 = sqrt(r136291);
        double r136293 = r136292 * r136289;
        double r136294 = r136293 + r136291;
        double r136295 = r136290 / r136294;
        return r136295;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.4

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

  7. Applied associate-*r/19.4

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

  8. Applied associate-/l/19.4

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

  9. Simplified19.4

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

  10. Taylor expanded around 0 0.7

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

  12. Applied associate-/r*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(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)))))