2isqrt (example 3.6)

Average Error: 19.6 → 0.3

Time: 17.4s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r102452 = 1.0;
        double r102453 = x;
        double r102454 = sqrt(r102453);
        double r102455 = r102452 / r102454;
        double r102456 = r102453 + r102452;
        double r102457 = sqrt(r102456);
        double r102458 = r102452 / r102457;
        double r102459 = r102455 - r102458;
        return r102459;
}

↓

double f(double x) {
        double r102460 = 1.0;
        double r102461 = x;
        double r102462 = sqrt(r102461);
        double r102463 = r102460 / r102462;
        double r102464 = 1.0;
        double r102465 = r102461 + r102464;
        double r102466 = sqrt(r102465);
        double r102467 = fma(r102466, r102462, r102465);
        double r102468 = r102460 / r102467;
        double r102469 = r102463 * r102468;
        return r102469;
}

Error

Bits error versus x

Target

Original	19.6
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.6

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

3. Applied frac-sub 19.6

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

5. Applied flip-- 19.5

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

6. Simplified 19.1

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

7. Simplified 19.1

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

8. Taylor expanded around 0 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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