2isqrt (example 3.6)

Average Error: 19.6 → 0.3

Time: 18.5s

Precision: 64

Math

\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]

↓

\[\frac{\frac{\sqrt{1}}{1}}{\sqrt{x}} \cdot \frac{\sqrt{1}}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x + 1\right)}\]

C

double f(double x) {
        double r88918 = 1.0;
        double r88919 = x;
        double r88920 = sqrt(r88919);
        double r88921 = r88918 / r88920;
        double r88922 = r88919 + r88918;
        double r88923 = sqrt(r88922);
        double r88924 = r88918 / r88923;
        double r88925 = r88921 - r88924;
        return r88925;
}

↓

double f(double x) {
        double r88926 = 1.0;
        double r88927 = sqrt(r88926);
        double r88928 = r88927 / r88926;
        double r88929 = x;
        double r88930 = sqrt(r88929);
        double r88931 = r88928 / r88930;
        double r88932 = r88929 + r88926;
        double r88933 = sqrt(r88932);
        double r88934 = fma(r88933, r88930, r88932);
        double r88935 = r88927 / r88934;
        double r88936 = r88931 * r88935;
        return r88936;
}

Error

Bits error versus x

Target

Original	19.6
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.6

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

3. Applied frac-sub 19.6

   \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
4. Using strategy rm

5. Applied flip-- 19.5

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

6. Simplified 19.1

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

7. Simplified 19.1

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

8. Taylor expanded around 0 0.4

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

10. Applied add-sqr-sqrt 0.4

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

11. Applied times-frac 0.4

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

12. Applied times-frac 0.4

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))))