Average Error: 47.0 → 0.1
Time: 11.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}\]
\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}
↓
\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}
double f(double i) {
        double r38303 = i;
        double r38304 = r38303 * r38303;
        double r38305 = r38304 * r38304;
        double r38306 = 2.0;
        double r38307 = r38306 * r38303;
        double r38308 = r38307 * r38307;
        double r38309 = r38305 / r38308;
        double r38310 = 1.0;
        double r38311 = r38308 - r38310;
        double r38312 = r38309 / r38311;
        return r38312;
}
↓
double f(double i) {
        double r38313 = 1.0;
        double r38314 = 2.0;
        double r38315 = r38313 / r38314;
        double r38316 = 1.0;
        double r38317 = sqrt(r38316);
        double r38318 = i;
        double r38319 = r38317 / r38318;
        double r38320 = r38314 + r38319;
        double r38321 = r38315 / r38320;
        double r38322 = r38314 - r38319;
        double r38323 = r38315 / r38322;
        double r38324 = r38321 * r38323;
        return r38324;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.0

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

  2. Simplified0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied times-frac0.4

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

  6. Applied difference-of-squares0.4

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

  7. Applied add-cube-cbrt0.4

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{2 \cdot 2}}{\left(2 + \frac{\sqrt{1}}{i}\right) \cdot \left(2 - \frac{\sqrt{1}}{i}\right)}\]

  8. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{2} \cdot \frac{\sqrt[3]{1}}{2}}}{\left(2 + \frac{\sqrt{1}}{i}\right) \cdot \left(2 - \frac{\sqrt{1}}{i}\right)}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{\sqrt[3]{1}}{2}}{2 - \frac{\sqrt{1}}{i}}}\]

  10. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}}} \cdot \frac{\frac{\sqrt[3]{1}}{2}}{2 - \frac{\sqrt{1}}{i}}\]

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +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)))