Average Error: 46.7 → 0.1
Time: 16.2s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{1}{2 - \frac{\sqrt{1}}{i}}}{2 \cdot 2}\]
\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{\frac{1}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{1}{2 - \frac{\sqrt{1}}{i}}}{2 \cdot 2}
double f(double i) {
        double r72494 = i;
        double r72495 = r72494 * r72494;
        double r72496 = r72495 * r72495;
        double r72497 = 2.0;
        double r72498 = r72497 * r72494;
        double r72499 = r72498 * r72498;
        double r72500 = r72496 / r72499;
        double r72501 = 1.0;
        double r72502 = r72499 - r72501;
        double r72503 = r72500 / r72502;
        return r72503;
}
↓
double f(double i) {
        double r72504 = 1.0;
        double r72505 = 2.0;
        double r72506 = 1.0;
        double r72507 = sqrt(r72506);
        double r72508 = i;
        double r72509 = r72507 / r72508;
        double r72510 = r72505 + r72509;
        double r72511 = r72504 / r72510;
        double r72512 = r72505 - r72509;
        double r72513 = r72504 / r72512;
        double r72514 = r72511 * r72513;
        double r72515 = r72505 * r72505;
        double r72516 = r72514 / r72515;
        return r72516;
}

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

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

  4. Applied associate-/r*0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied times-frac0.4

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

  8. Applied difference-of-squares0.4

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

  9. Applied associate-/r*0.1

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

  11. Applied div-inv0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))