2isqrt (example 3.6)

Average Error: 19.8 → 0.5

Time: 1.0m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r9219399 = 1.0;
        double r9219400 = x;
        double r9219401 = sqrt(r9219400);
        double r9219402 = r9219399 / r9219401;
        double r9219403 = r9219400 + r9219399;
        double r9219404 = sqrt(r9219403);
        double r9219405 = r9219399 / r9219404;
        double r9219406 = r9219402 - r9219405;
        return r9219406;
}

↓

double f(double x) {
        double r9219407 = 1.0;
        double r9219408 = x;
        double r9219409 = r9219408 + r9219407;
        double r9219410 = sqrt(r9219409);
        double r9219411 = sqrt(r9219408);
        double r9219412 = r9219410 + r9219411;
        double r9219413 = r9219407 / r9219412;
        double r9219414 = r9219407 / r9219410;
        double r9219415 = sqrt(r9219414);
        double r9219416 = r9219415 / r9219411;
        double r9219417 = r9219416 * r9219415;
        double r9219418 = r9219413 * r9219417;
        return r9219418;
}

Error

Bits error versus x

Target

Original	19.8
Target	0.6
Herbie	0.5

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

Derivation

1. Initial program 19.8

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

3. Applied frac-sub 19.7

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

4. Simplified 19.7

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

6. Applied flip-- 19.5

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

7. Applied associate-/l/ 19.5

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

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 *-un-lft-identity 0.4

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

15. Applied add-sqr-sqrt 0.5

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

16. Applied times-frac 0.5

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

17. Simplified 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019120 
(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)))))