Average Error: 19.8 → 0.5
Time: 6.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r148960 = 1.0;
        double r148961 = x;
        double r148962 = sqrt(r148961);
        double r148963 = r148960 / r148962;
        double r148964 = r148961 + r148960;
        double r148965 = sqrt(r148964);
        double r148966 = r148960 / r148965;
        double r148967 = r148963 - r148966;
        return r148967;
}
↓
double f(double x) {
        double r148968 = 1.0;
        double r148969 = sqrt(r148968);
        double r148970 = x;
        double r148971 = r148970 + r148968;
        double r148972 = sqrt(r148971);
        double r148973 = sqrt(r148970);
        double r148974 = r148972 + r148973;
        double r148975 = sqrt(r148974);
        double r148976 = r148969 / r148975;
        double r148977 = r148976 * r148976;
        double r148978 = r148968 * r148977;
        double r148979 = r148973 * r148972;
        double r148980 = r148978 / r148979;
        return r148980;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Simplified19.1

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

  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 add-sqr-sqrt0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied times-frac0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019344 
(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)))))