Average Error: 20.1 → 0.8
Time: 18.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot 1}{\left(\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x\right) \cdot \left(x + 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot 1}{\left(\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x\right) \cdot \left(x + 1\right)}
double f(double x) {
        double r75971 = 1.0;
        double r75972 = x;
        double r75973 = sqrt(r75972);
        double r75974 = r75971 / r75973;
        double r75975 = r75972 + r75971;
        double r75976 = sqrt(r75975);
        double r75977 = r75971 / r75976;
        double r75978 = r75974 - r75977;
        return r75978;
}
↓
double f(double x) {
        double r75979 = 1.0;
        double r75980 = r75979 * r75979;
        double r75981 = x;
        double r75982 = sqrt(r75981);
        double r75983 = r75979 / r75982;
        double r75984 = r75981 + r75979;
        double r75985 = sqrt(r75984);
        double r75986 = r75979 / r75985;
        double r75987 = r75983 + r75986;
        double r75988 = r75987 * r75981;
        double r75989 = r75988 * r75984;
        double r75990 = r75980 / r75989;
        return r75990;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.8
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied flip--20.1

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

  4. Simplified20.2

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

  6. Applied frac-sub19.6

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

  7. Applied associate-*r/19.6

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

  8. Applied associate-/l/19.5

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

  9. Taylor expanded around 0 5.7

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

  11. Applied associate-*r*0.8

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

  12. Final simplification0.8

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

Reproduce

herbie shell --seed 2019322 
(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)))))