2isqrt (example 3.6)

Average Error: 19.8 → 0.4

Time: 15.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r108854 = 1.0;
        double r108855 = x;
        double r108856 = sqrt(r108855);
        double r108857 = r108854 / r108856;
        double r108858 = r108855 + r108854;
        double r108859 = sqrt(r108858);
        double r108860 = r108854 / r108859;
        double r108861 = r108857 - r108860;
        return r108861;
}

↓

double f(double x) {
        double r108862 = 1.0;
        double r108863 = x;
        double r108864 = 1.0;
        double r108865 = r108863 + r108864;
        double r108866 = sqrt(r108865);
        double r108867 = sqrt(r108863);
        double r108868 = r108866 + r108867;
        double r108869 = r108862 / r108868;
        double r108870 = sqrt(r108869);
        double r108871 = r108870 / r108867;
        double r108872 = r108870 / r108866;
        double r108873 = r108871 * r108872;
        return r108873;
}

Error

Bits error versus x

Target

Original	19.8
Target	0.6
Herbie	0.4

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

Derivation

1. Initial program 19.8

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

3. Applied frac-sub 19.7

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

5. Applied flip-- 19.5

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

6. Simplified 19.1

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

7. Simplified 19.1

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

8. Taylor expanded around 0 0.4

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

10. Applied add-sqr-sqrt 0.4

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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)))))