Average Error: 45.4 → 0.2
Time: 21.8s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{i}{4 \cdot \left(\left(i \cdot 4 - \frac{\frac{1.0}{\sqrt{i}}}{\sqrt{i}}\right) + \left(\left(-\frac{\sqrt{1.0}}{\sqrt{i}}\right) + \frac{\sqrt{1.0}}{\sqrt{i}}\right) \cdot \frac{\sqrt{1.0}}{\sqrt{i}}\right)}\]
\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) - 1.0}
↓
\frac{i}{4 \cdot \left(\left(i \cdot 4 - \frac{\frac{1.0}{\sqrt{i}}}{\sqrt{i}}\right) + \left(\left(-\frac{\sqrt{1.0}}{\sqrt{i}}\right) + \frac{\sqrt{1.0}}{\sqrt{i}}\right) \cdot \frac{\sqrt{1.0}}{\sqrt{i}}\right)}
double f(double i) {
        double r3496857 = i;
        double r3496858 = r3496857 * r3496857;
        double r3496859 = r3496858 * r3496858;
        double r3496860 = 2.0;
        double r3496861 = r3496860 * r3496857;
        double r3496862 = r3496861 * r3496861;
        double r3496863 = r3496859 / r3496862;
        double r3496864 = 1.0;
        double r3496865 = r3496862 - r3496864;
        double r3496866 = r3496863 / r3496865;
        return r3496866;
}
↓
double f(double i) {
        double r3496867 = i;
        double r3496868 = 4.0;
        double r3496869 = r3496867 * r3496868;
        double r3496870 = 1.0;
        double r3496871 = sqrt(r3496867);
        double r3496872 = r3496870 / r3496871;
        double r3496873 = r3496872 / r3496871;
        double r3496874 = r3496869 - r3496873;
        double r3496875 = sqrt(r3496870);
        double r3496876 = r3496875 / r3496871;
        double r3496877 = -r3496876;
        double r3496878 = r3496877 + r3496876;
        double r3496879 = r3496878 * r3496876;
        double r3496880 = r3496874 + r3496879;
        double r3496881 = r3496868 * r3496880;
        double r3496882 = r3496867 / r3496881;
        return r3496882;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{i}{\left(4 \cdot i - \frac{1.0}{i}\right) \cdot 4}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

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

  5. Applied add-cube-cbrt0.2

    \[\leadsto \frac{i}{\left(4 \cdot i - \frac{\color{blue}{\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right) \cdot \sqrt[3]{1.0}}}{\sqrt{i} \cdot \sqrt{i}}\right) \cdot 4}\]

  6. Applied times-frac0.3

    \[\leadsto \frac{i}{\left(4 \cdot i - \color{blue}{\frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}} \cdot \frac{\sqrt[3]{1.0}}{\sqrt{i}}}\right) \cdot 4}\]

  7. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{i}{\left(\color{blue}{\sqrt{4 \cdot i} \cdot \sqrt{4 \cdot i}} - \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}} \cdot \frac{\sqrt[3]{1.0}}{\sqrt{i}}\right) \cdot 4}\]

  8. Applied prod-diff0.5

    \[\leadsto \frac{i}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{4 \cdot i}, \sqrt{4 \cdot i}, -\frac{\sqrt[3]{1.0}}{\sqrt{i}} \cdot \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1.0}}{\sqrt{i}}, \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}}, \frac{\sqrt[3]{1.0}}{\sqrt{i}} \cdot \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}}\right)\right)} \cdot 4}\]

  9. Simplified0.3

    \[\leadsto \frac{i}{\left(\color{blue}{\left(4 \cdot i - \frac{\frac{1.0}{\sqrt{i}}}{\sqrt{i}}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{1.0}}{\sqrt{i}}, \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}}, \frac{\sqrt[3]{1.0}}{\sqrt{i}} \cdot \frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt{i}}\right)\right) \cdot 4}\]

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +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)))