2isqrt (example 3.6)

Average Error: 19.4 → 0.4

Time: 15.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r131643 = 1.0;
        double r131644 = x;
        double r131645 = sqrt(r131644);
        double r131646 = r131643 / r131645;
        double r131647 = r131644 + r131643;
        double r131648 = sqrt(r131647);
        double r131649 = r131643 / r131648;
        double r131650 = r131646 - r131649;
        return r131650;
}

↓

double f(double x) {
        double r131651 = 1.0;
        double r131652 = x;
        double r131653 = sqrt(r131652);
        double r131654 = 1.0;
        double r131655 = r131652 + r131654;
        double r131656 = sqrt(r131655);
        double r131657 = r131653 + r131656;
        double r131658 = r131651 / r131657;
        double r131659 = sqrt(r131658);
        double r131660 = r131659 / r131653;
        double r131661 = r131659 / r131656;
        double r131662 = r131660 * r131661;
        return r131662;
}

Error

Bits error versus x

Target

Original	19.4
Target	0.8
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 18.9

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

7. Simplified 18.9

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

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-sqr-sqrt 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019212 +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)))))