2isqrt (example 3.6)

Average Error: 19.8 → 0.6

Time: 2.4m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r11009495 = 1.0;
        double r11009496 = x;
        double r11009497 = sqrt(r11009496);
        double r11009498 = r11009495 / r11009497;
        double r11009499 = r11009496 + r11009495;
        double r11009500 = sqrt(r11009499);
        double r11009501 = r11009495 / r11009500;
        double r11009502 = r11009498 - r11009501;
        return r11009502;
}

↓

double f(double x) {
        double r11009503 = 1.0;
        double r11009504 = x;
        double r11009505 = r11009504 + r11009503;
        double r11009506 = sqrt(r11009504);
        double r11009507 = sqrt(r11009505);
        double r11009508 = r11009504 * r11009507;
        double r11009509 = fma(r11009505, r11009506, r11009508);
        double r11009510 = r11009503 / r11009509;
        return r11009510;
}

Error

Bits error versus x

Target

Original	19.8
Target	0.6
Herbie	0.6

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

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 pow1 0.8

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

11. Applied pow1 0.8

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

12. Applied pow1 0.8

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

13. Applied pow-prod-down 0.8

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

14. Applied pow-prod-down 0.8

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

15. Simplified 0.6

    \[\leadsto \frac{1}{{\color{blue}{\left(\sqrt{x + 1} \cdot \mathsf{fma}\left(\left(\sqrt{x}\right), \left(\sqrt{x + 1}\right), x\right)\right)}}^{1}}\]
16. Using strategy rm

17. Applied *-un-lft-identity 0.6

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

18. Applied add-sqr-sqrt 0.6

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

19. Applied times-frac 0.6

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

20. Simplified 0.6

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

21. Simplified 0.6

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

22. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019121 +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)))))