2isqrt (example 3.6)

Average Error: 19.2 → 0.5

Time: 2.1m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r31202474 = 1.0;
        double r31202475 = x;
        double r31202476 = sqrt(r31202475);
        double r31202477 = r31202474 / r31202476;
        double r31202478 = r31202475 + r31202474;
        double r31202479 = sqrt(r31202478);
        double r31202480 = r31202474 / r31202479;
        double r31202481 = r31202477 - r31202480;
        return r31202481;
}

↓

double f(double x) {
        double r31202482 = 1.0;
        double r31202483 = x;
        double r31202484 = r31202483 + r31202482;
        double r31202485 = sqrt(r31202484);
        double r31202486 = sqrt(r31202483);
        double r31202487 = r31202485 + r31202486;
        double r31202488 = cbrt(r31202487);
        double r31202489 = r31202488 * r31202488;
        double r31202490 = r31202482 / r31202489;
        double r31202491 = sqrt(r31202490);
        double r31202492 = r31202485 * r31202486;
        double r31202493 = r31202482 / r31202487;
        double r31202494 = sqrt(r31202493);
        double r31202495 = r31202492 / r31202494;
        double r31202496 = r31202482 / r31202488;
        double r31202497 = sqrt(r31202496);
        double r31202498 = r31202495 / r31202497;
        double r31202499 = r31202491 / r31202498;
        return r31202499;
}

Error

Bits error versus x

Target

Original	19.2
Target	0.6
Herbie	0.5

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

Derivation

1. Initial program 19.2

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

3. Applied frac-sub 19.2

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

4. Simplified 19.2

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

6. Applied flip-- 19.0

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

7. Simplified 0.4

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

9. Applied add-sqr-sqrt 0.4

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

10. Applied associate-/l* 0.4

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

12. Applied add-cube-cbrt 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied times-frac 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied associate-/l* 0.5

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

17. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019125 
(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)))))