2isqrt (example 3.6)

Average Error: 19.5 → 0.4

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r133266 = 1.0;
        double r133267 = x;
        double r133268 = sqrt(r133267);
        double r133269 = r133266 / r133268;
        double r133270 = r133267 + r133266;
        double r133271 = sqrt(r133270);
        double r133272 = r133266 / r133271;
        double r133273 = r133269 - r133272;
        return r133273;
}

↓

double f(double x) {
        double r133274 = 1.0;
        double r133275 = sqrt(r133274);
        double r133276 = x;
        double r133277 = r133274 + r133276;
        double r133278 = sqrt(r133277);
        double r133279 = sqrt(r133278);
        double r133280 = r133275 / r133279;
        double r133281 = r133280 * r133280;
        double r133282 = sqrt(r133276);
        double r133283 = r133278 * r133282;
        double r133284 = r133276 + r133283;
        double r133285 = r133274 / r133284;
        double r133286 = r133281 * r133285;
        return r133286;
}

Error

Bits error versus x

Target

Original	19.5
Target	0.7
Herbie	0.4

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

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. Simplified 19.5

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

5. Simplified 19.5

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

7. Applied flip-- 19.3

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

8. Simplified 0.4

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

10. Applied times-frac 0.4

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

11. Simplified 0.4

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

12. Simplified 0.3

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

14. Applied add-sqr-sqrt 0.3

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

15. Applied sqrt-prod 0.4

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

16. Applied add-sqr-sqrt 0.4

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

17. Applied times-frac 0.4

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

18. Simplified 0.4

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))