Average Error: 45.3 → 0.1
Time: 18.4s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{1}{4}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\]
\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) - 1.0}
↓
\frac{\frac{1}{4}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}
double f(double i) {
        double r2611714 = i;
        double r2611715 = r2611714 * r2611714;
        double r2611716 = r2611715 * r2611715;
        double r2611717 = 2.0;
        double r2611718 = r2611717 * r2611714;
        double r2611719 = r2611718 * r2611718;
        double r2611720 = r2611716 / r2611719;
        double r2611721 = 1.0;
        double r2611722 = r2611719 - r2611721;
        double r2611723 = r2611720 / r2611722;
        return r2611723;
}
↓
double f(double i) {
        double r2611724 = 0.25;
        double r2611725 = 2.0;
        double r2611726 = 1.0;
        double r2611727 = sqrt(r2611726);
        double r2611728 = i;
        double r2611729 = r2611727 / r2611728;
        double r2611730 = r2611725 - r2611729;
        double r2611731 = r2611724 / r2611730;
        double r2611732 = 1.0;
        double r2611733 = r2611725 + r2611729;
        double r2611734 = r2611732 / r2611733;
        double r2611735 = r2611731 * r2611734;
        return r2611735;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.3

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{4}}{4 - \frac{1.0}{i \cdot i}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\frac{1}{4}}{4 - \frac{\color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{i \cdot i}}\]

  5. Applied times-frac0.5

    \[\leadsto \frac{\frac{1}{4}}{4 - \color{blue}{\frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}}}\]

  6. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\sqrt{4} \cdot \sqrt{4}} - \frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}}\]

  7. Applied difference-of-squares0.5

    \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)}}\]

  8. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{4}}}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{1}{\sqrt{4} + \frac{\sqrt{1.0}}{i}} \cdot \frac{\frac{1}{4}}{\sqrt{4} - \frac{\sqrt{1.0}}{i}}}\]

  10. Final simplification0.1

    \[\leadsto \frac{\frac{1}{4}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))