Average Error: 20.0 → 0.4
Time: 16.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r5385995 = 1.0;
        double r5385996 = x;
        double r5385997 = sqrt(r5385996);
        double r5385998 = r5385995 / r5385997;
        double r5385999 = r5385996 + r5385995;
        double r5386000 = sqrt(r5385999);
        double r5386001 = r5385995 / r5386000;
        double r5386002 = r5385998 - r5386001;
        return r5386002;
}
↓
double f(double x) {
        double r5386003 = 1.0;
        double r5386004 = x;
        double r5386005 = r5386004 + r5386003;
        double r5386006 = sqrt(r5386005);
        double r5386007 = sqrt(r5386004);
        double r5386008 = r5386006 + r5386007;
        double r5386009 = r5386003 / r5386008;
        double r5386010 = r5386003 * r5386009;
        double r5386011 = 1.0;
        double r5386012 = r5386006 * r5386007;
        double r5386013 = r5386011 / r5386012;
        double r5386014 = r5386010 * r5386013;
        return r5386014;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.8
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Simplified19.4

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

  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 div-inv0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))