2isqrt (example 3.6)

Average Error: 19.4 → 0.4

Time: 16.8s

Precision: 64

Math

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

↓

\[\frac{\frac{1}{1 \cdot \mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{x}}, \sqrt{x + 1}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

C

double f(double x) {
        double r102607 = 1.0;
        double r102608 = x;
        double r102609 = sqrt(r102608);
        double r102610 = r102607 / r102609;
        double r102611 = r102608 + r102607;
        double r102612 = sqrt(r102611);
        double r102613 = r102607 / r102612;
        double r102614 = r102610 - r102613;
        return r102614;
}

↓

double f(double x) {
        double r102615 = 1.0;
        double r102616 = x;
        double r102617 = cbrt(r102616);
        double r102618 = r102617 * r102617;
        double r102619 = sqrt(r102618);
        double r102620 = sqrt(r102617);
        double r102621 = r102616 + r102615;
        double r102622 = sqrt(r102621);
        double r102623 = fma(r102619, r102620, r102622);
        double r102624 = r102615 * r102623;
        double r102625 = r102615 / r102624;
        double r102626 = sqrt(r102616);
        double r102627 = r102626 * r102622;
        double r102628 = r102625 / r102627;
        return r102628;
}

Error

Bits error versus x

Target

Original	19.4
Target	0.6
Herbie	0.4

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

Derivation

1. Initial program 19.4

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

3. Applied frac-sub 19.4

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

5. Applied flip-- 19.2

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

6. Simplified 18.8

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

7. Simplified 18.8

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

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 add-cube-cbrt 0.4

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

11. Applied sqrt-prod 0.4

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

12. Applied fma-def 0.4

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

13. Final simplification 0.4

    \[\leadsto \frac{\frac{1}{1 \cdot \mathsf{fma}\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \sqrt{\sqrt[3]{x}}, \sqrt{x + 1}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

Reproduce

herbie shell --seed 2019303 +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)))))