2isqrt (example 3.6)

Average Error: 19.6 → 0.3

Time: 15.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r113670 = 1.0;
        double r113671 = x;
        double r113672 = sqrt(r113671);
        double r113673 = r113670 / r113672;
        double r113674 = r113671 + r113670;
        double r113675 = sqrt(r113674);
        double r113676 = r113670 / r113675;
        double r113677 = r113673 - r113676;
        return r113677;
}

↓

double f(double x) {
        double r113678 = 1.0;
        double r113679 = 1.0;
        double r113680 = r113678 / r113679;
        double r113681 = x;
        double r113682 = sqrt(r113681);
        double r113683 = r113680 / r113682;
        double r113684 = r113681 + r113679;
        double r113685 = sqrt(r113684);
        double r113686 = fma(r113685, r113682, r113684);
        double r113687 = r113679 / r113686;
        double r113688 = r113683 * r113687;
        return r113688;
}

Error

Bits error versus x

Target

Original	19.6
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 19.6

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

3. Applied frac-sub 19.6

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

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

7. Simplified 19.1

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

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 *-un-lft-identity 0.4

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

11. Applied times-frac 0.4

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

12. Applied times-frac 0.4

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019323 +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)))))