2isqrt (example 3.6)

Average Error: 19.5 → 0.3

Time: 18.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5371660 = 1.0;
        double r5371661 = x;
        double r5371662 = sqrt(r5371661);
        double r5371663 = r5371660 / r5371662;
        double r5371664 = r5371661 + r5371660;
        double r5371665 = sqrt(r5371664);
        double r5371666 = r5371660 / r5371665;
        double r5371667 = r5371663 - r5371666;
        return r5371667;
}

↓

double f(double x) {
        double r5371668 = 1.0;
        double r5371669 = x;
        double r5371670 = r5371669 + r5371668;
        double r5371671 = sqrt(r5371670);
        double r5371672 = r5371668 / r5371671;
        double r5371673 = sqrt(r5371669);
        double r5371674 = r5371673 * r5371671;
        double r5371675 = r5371669 + r5371674;
        double r5371676 = r5371672 / r5371675;
        return r5371676;
}

Error

Bits error versus x

Target

Original	19.5
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.5

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

3. Applied frac-sub 19.5

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

5. Applied flip-- 19.3

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

6. Simplified 18.9

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

7. Simplified 18.9

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

9. Applied associate-/r* 18.9

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

10. Simplified 0.3

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

12. Applied *-un-lft-identity 0.3

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

13. Applied sqrt-prod 0.3

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

14. Applied *-un-lft-identity 0.3

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

15. Applied add-cube-cbrt 0.3

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

16. Applied times-frac 0.3

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

17. Applied times-frac 0.3

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

18. Simplified 0.3

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

19. Simplified 0.3

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

20. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019158 
(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)))))