2isqrt (example 3.6)

Average Error: 20.2 → 0.4

Time: 19.4s

Precision: 64

Math

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

↓

\[\frac{\frac{1}{1 \cdot \mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

C

double f(double x) {
        double r119827 = 1.0;
        double r119828 = x;
        double r119829 = sqrt(r119828);
        double r119830 = r119827 / r119829;
        double r119831 = r119828 + r119827;
        double r119832 = sqrt(r119831);
        double r119833 = r119827 / r119832;
        double r119834 = r119830 - r119833;
        return r119834;
}

↓

double f(double x) {
        double r119835 = 1.0;
        double r119836 = x;
        double r119837 = r119836 + r119835;
        double r119838 = cbrt(r119837);
        double r119839 = r119838 * r119838;
        double r119840 = sqrt(r119839);
        double r119841 = sqrt(r119838);
        double r119842 = sqrt(r119836);
        double r119843 = fma(r119840, r119841, r119842);
        double r119844 = r119835 * r119843;
        double r119845 = r119835 / r119844;
        double r119846 = sqrt(r119837);
        double r119847 = r119846 * r119842;
        double r119848 = r119845 / r119847;
        return r119848;
}

Error

Bits error versus x

Target

Original	20.2
Target	0.7
Herbie	0.4

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

Derivation

1. Initial program 20.2

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

3. Applied frac-sub 20.2

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

4. Simplified 20.2

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

6. Applied flip-- 20.0

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

7. Simplified 19.6

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

8. Simplified 19.6

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

9. Taylor expanded around 0 0.4

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

11. Applied add-cube-cbrt 0.4

    \[\leadsto \frac{\frac{1}{1 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} + \sqrt{x}\right)}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

12. Applied sqrt-prod 0.4

    \[\leadsto \frac{\frac{1}{1 \cdot \left(\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} + \sqrt{x}\right)}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

13. Applied fma-def 0.4

    \[\leadsto \frac{\frac{1}{1 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

14. Final simplification 0.4

    \[\leadsto \frac{\frac{1}{1 \cdot \mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}}{\sqrt{x + 1} \cdot \sqrt{x}}\]

Reproduce

herbie shell --seed 2019194 +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)))))