Average Error: 46.6 → 0.1
Time: 1.5min
Precision: binary64
Cost: 2240
\[i > 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]
\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}
↓
\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}
(FPCore (i)
 :precision binary64
 (/
  (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i)))
  (- (* (* 2.0 i) (* 2.0 i)) 1.0)))
↓
(FPCore (i)
 :precision binary64
 (*
  (*
   (* (cbrt 1.0) (cbrt 1.0))
   (*
    (pow (cbrt (cbrt 1.0)) 4.0)
    (/ (cbrt (cbrt 1.0)) (/ (+ 2.0 (fabs (/ 1.0 i))) (cbrt (cbrt 1.0))))))
  (/ 0.25 (- 2.0 (fabs (/ 1.0 i))))))
double code(double i) {
	return (((i * i) * (i * i)) / ((2.0 * i) * (2.0 * i))) / (((2.0 * i) * (2.0 * i)) - 1.0);
}
↓
double code(double i) {
	return ((cbrt(1.0) * cbrt(1.0)) * (pow(cbrt(cbrt(1.0)), 4.0) * (cbrt(cbrt(1.0)) / ((2.0 + fabs(1.0 / i)) / cbrt(cbrt(1.0)))))) * (0.25 / (2.0 - fabs(1.0 / i)));
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy0.2
Cost2112
\[\frac{\frac{\sqrt[3]{0.25} \cdot \sqrt[3]{0.25}}{\sqrt{2} + \sqrt{\left|\frac{1}{i}\right|}}}{2 + \left|\frac{1}{i}\right|} \cdot \frac{\sqrt[3]{0.25}}{\sqrt{2} - \sqrt{\left|\frac{1}{i}\right|}}\]
Alternative 2
Accuracy0.9
Cost2944
\[\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{\sqrt[3]{1}}{\frac{\sqrt{2 + \left|\frac{1}{i}\right|}}{\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}}}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4}}{\sqrt{2 + \left|\frac{1}{i}\right|}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]
Alternative 3
Accuracy1.6
Cost2624
\[\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{2 + \left|\frac{1}{i}\right|} \cdot \sqrt[3]{2 + \left|\frac{1}{i}\right|}}}{\sqrt[3]{2 + \left|\frac{1}{i}\right|}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

Derivation

  1. Initial program 46.6

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{1}{i \cdot i}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary64_31690.5

    \[\leadsto \frac{0.25}{4 - \color{blue}{\sqrt{\frac{1}{i \cdot i}} \cdot \sqrt{\frac{1}{i \cdot i}}}}\]

  5. Applied add-sqr-sqrt_binary64_31690.5

    \[\leadsto \frac{0.25}{\color{blue}{\sqrt{4} \cdot \sqrt{4}} - \sqrt{\frac{1}{i \cdot i}} \cdot \sqrt{\frac{1}{i \cdot i}}}\]

  6. Applied difference-of-squares_binary64_31160.5

    \[\leadsto \frac{0.25}{\color{blue}{\left(\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}\right) \cdot \left(\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}\right)}}\]

  7. Applied *-un-lft-identity_binary64_31470.5

    \[\leadsto \frac{\color{blue}{1 \cdot 0.25}}{\left(\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}\right) \cdot \left(\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}\right)}\]

  8. Applied times-frac_binary64_31530.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{4} + \sqrt{\frac{1}{i \cdot i}}} \cdot \frac{0.25}{\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}}}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{2 + \left|\frac{1}{i}\right|}} \cdot \frac{0.25}{\sqrt{4} - \sqrt{\frac{1}{i \cdot i}}}\]

  10. Simplified0.1

    \[\leadsto \frac{1}{2 + \left|\frac{1}{i}\right|} \cdot \color{blue}{\frac{0.25}{2 - \left|\frac{1}{i}\right|}}\]
  11. Using strategy rm

  12. Applied *-un-lft-identity_binary64_31470.1

    \[\leadsto \frac{1}{\color{blue}{1 \cdot \left(2 + \left|\frac{1}{i}\right|\right)}} \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  13. Applied add-cube-cbrt_binary64_31820.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \left(2 + \left|\frac{1}{i}\right|\right)} \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  14. Applied times-frac_binary64_31530.1

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{2 + \left|\frac{1}{i}\right|}\right)} \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  15. Simplified0.1

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)} \cdot \frac{\sqrt[3]{1}}{2 + \left|\frac{1}{i}\right|}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]
  16. Using strategy rm

  17. Applied add-sqr-sqrt_binary64_31690.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}}{2 + \left|\frac{1}{i}\right|}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  18. Applied cbrt-prod_binary64_31780.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}}{2 + \left|\frac{1}{i}\right|}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  19. Applied associate-/l*_binary64_30920.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \color{blue}{\frac{\sqrt[3]{\sqrt{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt{1}}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  20. Simplified0.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{\sqrt{1}}}{\color{blue}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{1}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]
  21. Using strategy rm

  22. Applied add-cube-cbrt_binary64_31820.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{\sqrt{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}\right) \cdot \sqrt[3]{\sqrt[3]{1}}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  23. Applied *-un-lft-identity_binary64_31470.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{\sqrt{1}}}{\frac{\color{blue}{1 \cdot \left(2 + \left|\frac{1}{i}\right|\right)}}{\left(\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}\right) \cdot \sqrt[3]{\sqrt[3]{1}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  24. Applied times-frac_binary64_31530.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{\sqrt{1}}}{\color{blue}{\frac{1}{\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}} \cdot \frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  25. Applied add-cube-cbrt_binary64_31820.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{1}}}}}{\frac{1}{\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}} \cdot \frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  26. Applied times-frac_binary64_31530.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{\sqrt{1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1}}}}{\frac{1}{\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt{1}}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right)}\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  27. Simplified0.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt{1}}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  28. Simplified0.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4} \cdot \color{blue}{\frac{\sqrt[3]{\sqrt[3]{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

  29. Final simplification0.1

    \[\leadsto \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{1}}\right)}^{4} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\frac{2 + \left|\frac{1}{i}\right|}{\sqrt[3]{\sqrt[3]{1}}}}\right)\right) \cdot \frac{0.25}{2 - \left|\frac{1}{i}\right|}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))