Average Error: 45.4 → 0.1
Time: 18.3s
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 r1525571 = i;
        double r1525572 = r1525571 * r1525571;
        double r1525573 = r1525572 * r1525572;
        double r1525574 = 2.0;
        double r1525575 = r1525574 * r1525571;
        double r1525576 = r1525575 * r1525575;
        double r1525577 = r1525573 / r1525576;
        double r1525578 = 1.0;
        double r1525579 = r1525576 - r1525578;
        double r1525580 = r1525577 / r1525579;
        return r1525580;
}
↓
double f(double i) {
        double r1525581 = 0.25;
        double r1525582 = 2.0;
        double r1525583 = 1.0;
        double r1525584 = sqrt(r1525583);
        double r1525585 = i;
        double r1525586 = r1525584 / r1525585;
        double r1525587 = r1525582 - r1525586;
        double r1525588 = r1525581 / r1525587;
        double r1525589 = 1.0;
        double r1525590 = r1525582 + r1525586;
        double r1525591 = r1525589 / r1525590;
        double r1525592 = r1525588 * r1525591;
        return r1525592;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.4

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{4 - \frac{1.0}{i \cdot i}} \cdot \frac{1}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

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

  5. Applied times-frac0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied difference-of-squares0.4

    \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)}} \cdot \frac{1}{2}\]

  8. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{2}}}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)} \cdot \frac{1}{2}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{4} + \frac{\sqrt{1.0}}{i}} \cdot \frac{\frac{1}{2}}{\sqrt{4} - \frac{\sqrt{1.0}}{i}}\right)} \cdot \frac{1}{2}\]

  10. Applied associate-*l*0.1

    \[\leadsto \color{blue}{\frac{1}{\sqrt{4} + \frac{\sqrt{1.0}}{i}} \cdot \left(\frac{\frac{1}{2}}{\sqrt{4} - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2}\right)}\]

  11. Simplified0.1

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

  12. 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 2019153 
(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)))