Average Error: 20.3 → 20.3
Time: 33.6s
Precision: binary64
Cost: 1984
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \left(\sqrt[3]{1} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{x}} - \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{1 + x}}\right)\right)\right)\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \left(\sqrt[3]{1} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{x}} - \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{1 + x}}\right)\right)\right)
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (* (cbrt 1.0) (cbrt 1.0))
  (*
   (cbrt (* (cbrt 1.0) (cbrt 1.0)))
   (*
    (cbrt 1.0)
    (-
     (/ (cbrt (cbrt 1.0)) (sqrt x))
     (/ (cbrt (cbrt 1.0)) (sqrt (+ 1.0 x))))))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt(x + 1.0));
}
↓
double code(double x) {
	return (cbrt(1.0) * cbrt(1.0)) * (cbrt(cbrt(1.0) * cbrt(1.0)) * (cbrt(1.0) * ((cbrt(cbrt(1.0)) / sqrt(x)) - (cbrt(cbrt(1.0)) / sqrt(1.0 + x)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target0.7
Herbie20.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Alternatives

Alternative 1
Accuracy26.2
Cost2496
\[\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{x}} - \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{\sqrt{1 + x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{\sqrt{1 + x}}}\right)\right)\]
Alternative 2
Accuracy28.9
Cost2496
\[\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt[3]{\sqrt{x}}} - \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt{1 + x}}\right)\right)\]

Derivation

  1. Initial program 20.3

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

  3. Applied *-un-lft-identity_binary64_417020.3

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

  4. Applied sqrt-prod_binary64_418620.3

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

  5. Applied add-cube-cbrt_binary64_420520.3

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

  6. Applied times-frac_binary64_417620.3

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

  7. Applied *-un-lft-identity_binary64_417020.3

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

  8. Applied sqrt-prod_binary64_418620.3

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

  9. Applied add-cube-cbrt_binary64_420520.3

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

  10. Applied times-frac_binary64_417620.3

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

  11. Applied distribute-lft-out--_binary64_412220.3

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

  12. Simplified20.3

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

  14. Applied *-un-lft-identity_binary64_417020.3

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

  15. Applied add-cube-cbrt_binary64_420520.3

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

  16. Applied cbrt-prod_binary64_420120.3

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

  17. Applied times-frac_binary64_417620.3

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

  18. Applied *-un-lft-identity_binary64_417020.3

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

  19. Applied add-cube-cbrt_binary64_420520.3

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

  20. Applied cbrt-prod_binary64_420120.3

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

  21. Applied times-frac_binary64_417620.3

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

  22. Applied distribute-lft-out--_binary64_412220.3

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

  24. Applied *-un-lft-identity_binary64_417020.3

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

  25. Applied *-un-lft-identity_binary64_417020.3

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

  26. Applied cbrt-prod_binary64_420120.3

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

  27. Applied times-frac_binary64_417620.3

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

  28. Applied *-un-lft-identity_binary64_417020.3

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

  29. Applied *-un-lft-identity_binary64_417020.3

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

  30. Applied cbrt-prod_binary64_420120.3

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

  31. Applied times-frac_binary64_417620.3

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

  32. Applied distribute-lft-out--_binary64_412220.3

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

  33. Final simplification20.3

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

Reproduce

herbie shell --seed 2020338 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))