2isqrt (example 3.6)

Average Error: 19.4 → 0.4

Time: 17.7s

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 r70283 = 1.0;
        double r70284 = x;
        double r70285 = sqrt(r70284);
        double r70286 = r70283 / r70285;
        double r70287 = r70284 + r70283;
        double r70288 = sqrt(r70287);
        double r70289 = r70283 / r70288;
        double r70290 = r70286 - r70289;
        return r70290;
}

↓

double f(double x) {
        double r70291 = 1.0;
        double r70292 = x;
        double r70293 = cbrt(r70292);
        double r70294 = r70293 * r70293;
        double r70295 = sqrt(r70294);
        double r70296 = sqrt(r70293);
        double r70297 = r70292 + r70291;
        double r70298 = sqrt(r70297);
        double r70299 = fma(r70295, r70296, r70298);
        double r70300 = r70291 * r70299;
        double r70301 = r70291 / r70300;
        double r70302 = sqrt(r70292);
        double r70303 = r70302 * r70298;
        double r70304 = r70301 / r70303;
        return r70304;
}

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)))))