Average Error: 46.4 → 0.1
Time: 3.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{1}{2 \cdot i + \sqrt{1}} \cdot \left(\frac{i}{2 \cdot i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}\right)\]
\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{1}{2 \cdot i + \sqrt{1}} \cdot \left(\frac{i}{2 \cdot i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}\right)
double f(double i) {
        double r71477 = i;
        double r71478 = r71477 * r71477;
        double r71479 = r71478 * r71478;
        double r71480 = 2.0;
        double r71481 = r71480 * r71477;
        double r71482 = r71481 * r71481;
        double r71483 = r71479 / r71482;
        double r71484 = 1.0;
        double r71485 = r71482 - r71484;
        double r71486 = r71483 / r71485;
        return r71486;
}
↓
double f(double i) {
        double r71487 = 1.0;
        double r71488 = 2.0;
        double r71489 = i;
        double r71490 = r71488 * r71489;
        double r71491 = 1.0;
        double r71492 = sqrt(r71491);
        double r71493 = r71490 + r71492;
        double r71494 = r71487 / r71493;
        double r71495 = r71490 - r71492;
        double r71496 = r71489 / r71495;
        double r71497 = r71488 * r71488;
        double r71498 = r71489 / r71497;
        double r71499 = r71496 * r71498;
        double r71500 = r71494 * r71499;
        return r71500;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.4

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

  2. Simplified16.3

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

  4. Applied times-frac15.8

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

  6. Applied add-sqr-sqrt15.8

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

  7. Applied difference-of-squares15.8

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

  8. Applied *-un-lft-identity15.8

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

  9. Applied times-frac0.1

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

  10. Applied associate-*l*0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020059 +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)))