2isqrt (example 3.6)

Average Error: 19.9 → 0.3

Time: 14.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r101953 = 1.0;
        double r101954 = x;
        double r101955 = sqrt(r101954);
        double r101956 = r101953 / r101955;
        double r101957 = r101954 + r101953;
        double r101958 = sqrt(r101957);
        double r101959 = r101953 / r101958;
        double r101960 = r101956 - r101959;
        return r101960;
}

↓

double f(double x) {
        double r101961 = 1.0;
        double r101962 = sqrt(r101961);
        double r101963 = r101962 / r101961;
        double r101964 = x;
        double r101965 = sqrt(r101964);
        double r101966 = r101963 / r101965;
        double r101967 = r101964 + r101961;
        double r101968 = sqrt(r101967);
        double r101969 = fma(r101968, r101965, r101967);
        double r101970 = r101962 / r101969;
        double r101971 = r101966 * r101970;
        return r101971;
}

Error

Bits error versus x

Target

Original	19.9
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.9

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

3. Applied frac-sub 19.9

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

5. Applied flip-- 19.7

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

6. Simplified 19.3

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

7. Simplified 19.3

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

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

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

11. Applied times-frac 0.4

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

12. Applied times-frac 0.4

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019304 +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)))))