2isqrt (example 3.6)

Average Error: 19.3 → 0.3

Time: 16.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r3577491 = 1.0;
        double r3577492 = x;
        double r3577493 = sqrt(r3577492);
        double r3577494 = r3577491 / r3577493;
        double r3577495 = r3577492 + r3577491;
        double r3577496 = sqrt(r3577495);
        double r3577497 = r3577491 / r3577496;
        double r3577498 = r3577494 - r3577497;
        return r3577498;
}

↓

double f(double x) {
        double r3577499 = 1.0;
        double r3577500 = x;
        double r3577501 = r3577500 + r3577499;
        double r3577502 = sqrt(r3577500);
        double r3577503 = sqrt(r3577501);
        double r3577504 = r3577502 * r3577503;
        double r3577505 = r3577501 - r3577504;
        double r3577506 = r3577500 + r3577505;
        double r3577507 = r3577499 / r3577506;
        double r3577508 = r3577502 + r3577503;
        double r3577509 = r3577499 / r3577508;
        double r3577510 = r3577507 * r3577509;
        double r3577511 = r3577502 * r3577502;
        double r3577512 = r3577511 - r3577504;
        double r3577513 = r3577503 * r3577503;
        double r3577514 = r3577512 + r3577513;
        double r3577515 = r3577510 * r3577514;
        double r3577516 = r3577515 / r3577504;
        return r3577516;
}

Error

Bits error versus x

Target

Original	19.3
Target	0.6
Herbie	0.3

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

Derivation

1. Initial program 19.3

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

3. Applied frac-sub 19.3

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

4. Simplified 19.3

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

6. Applied flip-- 19.1

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

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 flip3-+ 0.8

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

10. Applied associate-/r/ 0.8

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

11. Simplified 0.7

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

13. Applied add-sqr-sqrt 0.7

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

14. Applied pow3 0.7

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

15. Applied add-sqr-sqrt 0.8

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

16. Applied pow3 0.8

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

17. Applied sum-cubes 0.8

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

18. Applied add-cube-cbrt 0.8

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

19. Applied times-frac 0.4

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

20. Simplified 0.3

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

21. Simplified 0.3

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

22. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019133 
(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)))))