Average Error: 54.1 → 37.2
Time: 16.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3.7225611126883413 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}
↓
\begin{array}{l}
\mathbf{if}\;i \le 3.7225611126883413 \cdot 10^{87}:\\
\;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\

\mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\
\;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta, double i) {
        double r149798 = i;
        double r149799 = alpha;
        double r149800 = beta;
        double r149801 = r149799 + r149800;
        double r149802 = r149801 + r149798;
        double r149803 = r149798 * r149802;
        double r149804 = r149800 * r149799;
        double r149805 = r149804 + r149803;
        double r149806 = r149803 * r149805;
        double r149807 = 2.0;
        double r149808 = r149807 * r149798;
        double r149809 = r149801 + r149808;
        double r149810 = r149809 * r149809;
        double r149811 = r149806 / r149810;
        double r149812 = 1.0;
        double r149813 = r149810 - r149812;
        double r149814 = r149811 / r149813;
        return r149814;
}
↓
double f(double alpha, double beta, double i) {
        double r149815 = i;
        double r149816 = 3.7225611126883413e+87;
        bool r149817 = r149815 <= r149816;
        double r149818 = alpha;
        double r149819 = beta;
        double r149820 = r149818 + r149819;
        double r149821 = 2.0;
        double r149822 = r149821 * r149815;
        double r149823 = r149820 + r149822;
        double r149824 = r149820 + r149815;
        double r149825 = r149823 / r149824;
        double r149826 = r149815 / r149825;
        double r149827 = 1.0;
        double r149828 = sqrt(r149827);
        double r149829 = r149823 + r149828;
        double r149830 = r149826 / r149829;
        double r149831 = r149819 * r149818;
        double r149832 = r149815 * r149824;
        double r149833 = r149831 + r149832;
        double r149834 = sqrt(r149833);
        double r149835 = r149823 / r149834;
        double r149836 = r149834 / r149835;
        double r149837 = r149830 * r149836;
        double r149838 = r149823 - r149828;
        double r149839 = r149837 / r149838;
        double r149840 = 6.726192858084654e+153;
        bool r149841 = r149815 <= r149840;
        double r149842 = 0.25;
        double r149843 = 2.0;
        double r149844 = pow(r149815, r149843);
        double r149845 = r149842 * r149844;
        double r149846 = r149823 * r149823;
        double r149847 = r149846 - r149827;
        double r149848 = r149845 / r149847;
        double r149849 = 0.0;
        double r149850 = r149841 ? r149848 : r149849;
        double r149851 = r149817 ? r149839 : r149850;
        return r149851;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if i < 3.7225611126883413e+87

    1. Initial program 29.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt29.1

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

    4. Applied difference-of-squares29.1

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

    5. Applied times-frac12.0

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

    6. Applied times-frac7.6

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\]
    7. Using strategy rm

    8. Applied associate-*r/7.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt7.6

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

    11. Applied associate-/l*7.6

      \[\leadsto \frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \color{blue}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
    12. Using strategy rm

    13. Applied associate-/l*7.6

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

    if 3.7225611126883413e+87 < i < 6.726192858084654e+153

    1. Initial program 64.0

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

    2. Taylor expanded around inf 18.4

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

    if 6.726192858084654e+153 < i

    1. Initial program 64.0

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

    2. Taylor expanded around inf 61.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification37.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.7225611126883413 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))