2isqrt (example 3.6)

Average Error: 19.6 → 0.4

Time: 20.0s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2364587 = 1.0;
        double r2364588 = x;
        double r2364589 = sqrt(r2364588);
        double r2364590 = r2364587 / r2364589;
        double r2364591 = r2364588 + r2364587;
        double r2364592 = sqrt(r2364591);
        double r2364593 = r2364587 / r2364592;
        double r2364594 = r2364590 - r2364593;
        return r2364594;
}

↓

double f(double x) {
        double r2364595 = 1.0;
        double r2364596 = x;
        double r2364597 = r2364596 + r2364595;
        double r2364598 = sqrt(r2364597);
        double r2364599 = sqrt(r2364596);
        double r2364600 = r2364598 + r2364599;
        double r2364601 = r2364595 / r2364600;
        double r2364602 = sqrt(r2364598);
        double r2364603 = r2364601 / r2364602;
        double r2364604 = r2364603 / r2364602;
        double r2364605 = r2364595 / r2364599;
        double r2364606 = r2364604 * r2364605;
        return r2364606;
}

Error

Bits error versus x

Target

Original	19.6
Target	0.7
Herbie	0.4

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

Derivation

1. Initial program 19.6

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

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

6. Applied flip-- 19.3

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

7. Simplified 18.9

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

9. Applied div-inv 18.9

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

10. Applied times-frac 18.9

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

11. Simplified 0.4

    \[\leadsto \color{blue}{\frac{1 + 0}{\sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}\]
12. Using strategy rm

13. Applied add-sqr-sqrt 0.5

    \[\leadsto \frac{1 + 0}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}}\]

14. Applied associate-/r* 0.4

    \[\leadsto \frac{1 + 0}{\sqrt{x}} \cdot \color{blue}{\frac{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{\sqrt{x + 1}}}}{\sqrt{\sqrt{x + 1}}}}\]

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019153 
(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)))))