2isqrt (example 3.6)

Average Error: 20.1 → 0.3

Time: 1.9m

Precision: 64

Math

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

↓

\[\frac{\frac{1}{\sqrt{x + 1}}}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x + 1}, x\right)}\]

C

double f(double x) {
        double r3406096 = 1.0;
        double r3406097 = x;
        double r3406098 = sqrt(r3406097);
        double r3406099 = r3406096 / r3406098;
        double r3406100 = r3406097 + r3406096;
        double r3406101 = sqrt(r3406100);
        double r3406102 = r3406096 / r3406101;
        double r3406103 = r3406099 - r3406102;
        return r3406103;
}

↓

double f(double x) {
        double r3406104 = 1.0;
        double r3406105 = x;
        double r3406106 = r3406105 + r3406104;
        double r3406107 = sqrt(r3406106);
        double r3406108 = r3406104 / r3406107;
        double r3406109 = sqrt(r3406105);
        double r3406110 = fma(r3406109, r3406107, r3406105);
        double r3406111 = r3406108 / r3406110;
        return r3406111;
}

Error

Bits error versus x

Target

Original	20.1
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 20.1

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

3. Applied frac-sub 20.0

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

4. Simplified 20.0

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

6. Applied flip-- 19.8

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

7. Simplified 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied associate-/l* 0.8

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

11. Simplified 0.6

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

13. Applied *-un-lft-identity 0.6

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

14. Applied add-sqr-sqrt 0.6

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

15. Applied times-frac 0.6

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

16. Simplified 0.6

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

17. Simplified 0.3

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

18. Final simplification 0.3

    \[\leadsto \frac{\frac{1}{\sqrt{x + 1}}}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x + 1}, x\right)}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))))