Average Error: 46.7 → 0.1
Time: 44.7s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\sqrt{\frac{\frac{1}{2}}{2}}}{2 + \frac{\sqrt{1}}{i}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \frac{\sqrt{\frac{\sqrt{\frac{1}{2}}}{2}}}{2 - \frac{\sqrt{1}}{i}}\right)\]
\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}
↓
\frac{\sqrt{\frac{\frac{1}{2}}{2}}}{2 + \frac{\sqrt{1}}{i}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \frac{\sqrt{\frac{\sqrt{\frac{1}{2}}}{2}}}{2 - \frac{\sqrt{1}}{i}}\right)
double f(double i) {
        double r98205 = i;
        double r98206 = r98205 * r98205;
        double r98207 = r98206 * r98206;
        double r98208 = 2.0;
        double r98209 = r98208 * r98205;
        double r98210 = r98209 * r98209;
        double r98211 = r98207 / r98210;
        double r98212 = 1.0;
        double r98213 = r98210 - r98212;
        double r98214 = r98211 / r98213;
        return r98214;
}
↓
double f(double i) {
        double r98215 = 1.0;
        double r98216 = 2.0;
        double r98217 = r98215 / r98216;
        double r98218 = r98217 / r98216;
        double r98219 = sqrt(r98218);
        double r98220 = 1.0;
        double r98221 = sqrt(r98220);
        double r98222 = i;
        double r98223 = r98221 / r98222;
        double r98224 = r98216 + r98223;
        double r98225 = r98219 / r98224;
        double r98226 = sqrt(r98217);
        double r98227 = sqrt(r98226);
        double r98228 = r98226 / r98216;
        double r98229 = sqrt(r98228);
        double r98230 = r98216 - r98223;
        double r98231 = r98229 / r98230;
        double r98232 = r98227 * r98231;
        double r98233 = r98225 * r98232;
        return r98233;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.7

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{2}}{2 \cdot 2 - \frac{1}{i \cdot i}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{\frac{\frac{1}{2}}{2}}{2 \cdot 2 - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{i \cdot i}}\]

  5. Applied times-frac0.5

    \[\leadsto \frac{\frac{\frac{1}{2}}{2}}{2 \cdot 2 - \color{blue}{\frac{\sqrt{1}}{i} \cdot \frac{\sqrt{1}}{i}}}\]

  6. Applied difference-of-squares0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{2}}{2}}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{2}}}{2 - \frac{\sqrt{1}}{i}}}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.1

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

  11. Applied *-un-lft-identity0.1

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

  12. Applied add-sqr-sqrt0.1

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

  13. Applied times-frac0.1

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

  14. Applied sqrt-prod0.1

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

  15. Applied times-frac0.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019199 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))