2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 9.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[{\left(\frac{1 \cdot 1}{\left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right) + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)}\right)}^{\frac{1}{2}}\]

C

double f(double x) {
        double r189078 = x;
        double r189079 = 1.0;
        double r189080 = r189078 + r189079;
        double r189081 = sqrt(r189080);
        double r189082 = sqrt(r189078);
        double r189083 = r189081 - r189082;
        return r189083;
}

↓

double f(double x) {
        double r189084 = 1.0;
        double r189085 = r189084 * r189084;
        double r189086 = x;
        double r189087 = r189086 + r189084;
        double r189088 = sqrt(r189087);
        double r189089 = sqrt(r189086);
        double r189090 = fma(r189088, r189089, r189086);
        double r189091 = r189084 + r189090;
        double r189092 = r189091 + r189090;
        double r189093 = r189085 / r189092;
        double r189094 = 0.5;
        double r189095 = pow(r189093, r189094);
        return r189095;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.5

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

3. Applied flip-- 29.4

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

4. Simplified 0.2

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

6. Applied add-sqr-sqrt 0.3

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

7. Simplified 0.3

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

8. Simplified 0.3

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

10. Applied pow1/2 0.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}}\]

11. Applied pow1/2 0.3

    \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}\]

12. Applied pow-prod-down 0.2

    \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}}\]

13. Simplified 0.3

    \[\leadsto {\color{blue}{\left(\frac{1 \cdot 1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)}}^{\frac{1}{2}}\]
14. Using strategy rm

15. Applied distribute-lft-in 0.2

    \[\leadsto {\left(\frac{1 \cdot 1}{\color{blue}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \sqrt{x + 1} + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \sqrt{x}}}\right)}^{\frac{1}{2}}\]

16. Simplified 0.2

    \[\leadsto {\left(\frac{1 \cdot 1}{\color{blue}{\left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right)} + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \sqrt{x}}\right)}^{\frac{1}{2}}\]

17. Simplified 0.2

    \[\leadsto {\left(\frac{1 \cdot 1}{\left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right) + \color{blue}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)}}\right)}^{\frac{1}{2}}\]

18. Final simplification 0.2

    \[\leadsto {\left(\frac{1 \cdot 1}{\left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right) + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)}\right)}^{\frac{1}{2}}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))