Average Error: 24.5 → 11.7
Time: 28.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.401252237322337023611530060789711214614 \cdot 10^{145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}}, \beta + \alpha, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right) + \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \beta}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6.401252237322337023611530060789711214614 \cdot 10^{145}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}}, \beta + \alpha, 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right) + \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \beta}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r5063780 = alpha;
        double r5063781 = beta;
        double r5063782 = r5063780 + r5063781;
        double r5063783 = r5063781 - r5063780;
        double r5063784 = r5063782 * r5063783;
        double r5063785 = 2.0;
        double r5063786 = i;
        double r5063787 = r5063785 * r5063786;
        double r5063788 = r5063782 + r5063787;
        double r5063789 = r5063784 / r5063788;
        double r5063790 = r5063788 + r5063785;
        double r5063791 = r5063789 / r5063790;
        double r5063792 = 1.0;
        double r5063793 = r5063791 + r5063792;
        double r5063794 = r5063793 / r5063785;
        return r5063794;
}
↓
double f(double alpha, double beta, double i) {
        double r5063795 = alpha;
        double r5063796 = 6.401252237322337e+145;
        bool r5063797 = r5063795 <= r5063796;
        double r5063798 = beta;
        double r5063799 = r5063798 - r5063795;
        double r5063800 = cbrt(r5063799);
        double r5063801 = r5063800 * r5063800;
        double r5063802 = 2.0;
        double r5063803 = i;
        double r5063804 = r5063798 + r5063795;
        double r5063805 = fma(r5063802, r5063803, r5063804);
        double r5063806 = r5063805 + r5063802;
        double r5063807 = cbrt(r5063806);
        double r5063808 = r5063807 * r5063807;
        double r5063809 = r5063801 / r5063808;
        double r5063810 = r5063800 / r5063805;
        double r5063811 = r5063810 / r5063807;
        double r5063812 = r5063809 * r5063811;
        double r5063813 = 1.0;
        double r5063814 = fma(r5063812, r5063804, r5063813);
        double r5063815 = r5063814 / r5063802;
        double r5063816 = 8.0;
        double r5063817 = r5063795 * r5063795;
        double r5063818 = r5063817 * r5063795;
        double r5063819 = r5063816 / r5063818;
        double r5063820 = r5063802 / r5063795;
        double r5063821 = r5063819 + r5063820;
        double r5063822 = 4.0;
        double r5063823 = r5063822 / r5063817;
        double r5063824 = r5063821 - r5063823;
        double r5063825 = r5063799 / r5063805;
        double r5063826 = r5063825 / r5063806;
        double r5063827 = r5063826 * r5063798;
        double r5063828 = r5063824 + r5063827;
        double r5063829 = r5063828 / r5063802;
        double r5063830 = r5063797 ? r5063815 : r5063829;
        return r5063830;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.401252237322337e+145

    1. Initial program 16.3

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

    2. Simplified5.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, \beta + \alpha, 1\right)}{2}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt5.6

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

    5. Applied *-un-lft-identity5.6

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

    6. Applied add-cube-cbrt5.5

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

    7. Applied times-frac5.5

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

    8. Applied times-frac5.4

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

    if 6.401252237322337e+145 < alpha

    1. Initial program 63.3

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

    2. Simplified47.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, \beta + \alpha, 1\right)}{2}}\]
    3. Using strategy rm

    4. Applied fma-udef46.6

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

    6. Applied distribute-rgt-in46.6

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

    7. Applied associate-+l+46.7

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

    8. Taylor expanded around inf 41.2

      \[\leadsto \frac{\beta \cdot \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} + \color{blue}{\left(\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]

    9. Simplified41.2

      \[\leadsto \frac{\beta \cdot \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{2 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} + \color{blue}{\left(\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.401252237322337023611530060789711214614 \cdot 10^{145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\sqrt[3]{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2}}, \beta + \alpha, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right) + \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2} \cdot \beta}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))