Average Error: 46.9 → 0.1
Time: 48.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{i}{2 \cdot i + \sqrt{1}} \cdot \frac{\frac{1}{2}}{\frac{2 \cdot i - \sqrt{1}}{\frac{i}{2}}}\]
\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{i}{2 \cdot i + \sqrt{1}} \cdot \frac{\frac{1}{2}}{\frac{2 \cdot i - \sqrt{1}}{\frac{i}{2}}}
double f(double i) {
        double r8511451 = i;
        double r8511452 = r8511451 * r8511451;
        double r8511453 = r8511452 * r8511452;
        double r8511454 = 2.0;
        double r8511455 = r8511454 * r8511451;
        double r8511456 = r8511455 * r8511455;
        double r8511457 = r8511453 / r8511456;
        double r8511458 = 1.0;
        double r8511459 = r8511456 - r8511458;
        double r8511460 = r8511457 / r8511459;
        return r8511460;
}
↓
double f(double i) {
        double r8511461 = i;
        double r8511462 = 2.0;
        double r8511463 = r8511462 * r8511461;
        double r8511464 = 1.0;
        double r8511465 = sqrt(r8511464);
        double r8511466 = r8511463 + r8511465;
        double r8511467 = r8511461 / r8511466;
        double r8511468 = 1.0;
        double r8511469 = r8511468 / r8511462;
        double r8511470 = r8511463 - r8511465;
        double r8511471 = r8511461 / r8511462;
        double r8511472 = r8511470 / r8511471;
        double r8511473 = r8511469 / r8511472;
        double r8511474 = r8511467 * r8511473;
        return r8511474;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.9

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

  2. Simplified15.8

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

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

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

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

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

  6. Applied times-frac15.8

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

  7. Applied add-sqr-sqrt15.8

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

  8. Applied difference-of-squares15.8

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

  9. Applied times-frac0.2

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

  10. Applied div-inv0.2

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

  11. Applied times-frac0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019173 
(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)))