2isqrt (example 3.6)

Average Error: 19.8 → 0.3

Time: 3.2m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r18897821 = 1.0;
        double r18897822 = x;
        double r18897823 = sqrt(r18897822);
        double r18897824 = r18897821 / r18897823;
        double r18897825 = r18897822 + r18897821;
        double r18897826 = sqrt(r18897825);
        double r18897827 = r18897821 / r18897826;
        double r18897828 = r18897824 - r18897827;
        return r18897828;
}

↓

double f(double x) {
        double r18897829 = 1.0;
        double r18897830 = x;
        double r18897831 = sqrt(r18897830);
        double r18897832 = r18897830 + r18897829;
        double r18897833 = sqrt(r18897832);
        double r18897834 = fma(r18897831, r18897833, r18897832);
        double r18897835 = r18897829 / r18897834;
        double r18897836 = r18897835 / r18897831;
        return r18897836;
}

Error

Bits error versus x

Target

Original	19.8
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.8

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

3. Applied frac-sub 19.8

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

4. Simplified 19.8

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

6. Applied flip-- 19.6

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

7. Applied associate-/l/ 19.6

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

8. Simplified 0.8

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

10. Applied add-cube-cbrt 0.8

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]

11. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x + 1} + \sqrt{x}}}\]

12. Simplified 0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x + 1} + \sqrt{x}}\]
13. Using strategy rm

14. Applied associate-*l/ 0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x + 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x}}}\]

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))