Average Error: 24.5 → 11.5
Time: 31.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, 1 \cdot 1, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\right)\right)}{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) + \left(1 \cdot 1 - \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(1, 1 \cdot 1, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\right)\right)}{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) + \left(1 \cdot 1 - \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot 1\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r9285781 = alpha;
        double r9285782 = beta;
        double r9285783 = r9285781 + r9285782;
        double r9285784 = r9285782 - r9285781;
        double r9285785 = r9285783 * r9285784;
        double r9285786 = 2.0;
        double r9285787 = i;
        double r9285788 = r9285786 * r9285787;
        double r9285789 = r9285783 + r9285788;
        double r9285790 = r9285785 / r9285789;
        double r9285791 = r9285789 + r9285786;
        double r9285792 = r9285790 / r9285791;
        double r9285793 = 1.0;
        double r9285794 = r9285792 + r9285793;
        double r9285795 = r9285794 / r9285786;
        return r9285795;
}
↓
double f(double alpha, double beta, double i) {
        double r9285796 = alpha;
        double r9285797 = 5.3206985442429606e+193;
        bool r9285798 = r9285796 <= r9285797;
        double r9285799 = 1.0;
        double r9285800 = r9285799 * r9285799;
        double r9285801 = beta;
        double r9285802 = r9285801 - r9285796;
        double r9285803 = 2.0;
        double r9285804 = i;
        double r9285805 = r9285796 + r9285801;
        double r9285806 = fma(r9285803, r9285804, r9285805);
        double r9285807 = r9285802 / r9285806;
        double r9285808 = r9285803 + r9285806;
        double r9285809 = r9285805 / r9285808;
        double r9285810 = r9285807 * r9285809;
        double r9285811 = r9285810 * r9285810;
        double r9285812 = r9285810 * r9285811;
        double r9285813 = fma(r9285799, r9285800, r9285812);
        double r9285814 = r9285810 * r9285799;
        double r9285815 = r9285800 - r9285814;
        double r9285816 = r9285811 + r9285815;
        double r9285817 = r9285813 / r9285816;
        double r9285818 = r9285817 / r9285803;
        double r9285819 = 8.0;
        double r9285820 = r9285796 * r9285796;
        double r9285821 = r9285820 * r9285796;
        double r9285822 = r9285819 / r9285821;
        double r9285823 = 4.0;
        double r9285824 = r9285823 / r9285820;
        double r9285825 = r9285822 - r9285824;
        double r9285826 = r9285803 / r9285796;
        double r9285827 = r9285825 + r9285826;
        double r9285828 = r9285827 / r9285803;
        double r9285829 = r9285798 ? r9285818 : r9285828;
        return r9285829;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.3206985442429606e+193

    1. Initial program 18.8

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

    2. Simplified7.5

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

    4. Applied add-log-exp7.5

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

    6. Applied fma-udef7.5

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

    7. Applied exp-sum7.5

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

    8. Applied log-prod7.5

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

    9. Simplified7.5

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

    10. Simplified7.5

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

    12. Applied flip3-+7.5

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

    13. Simplified7.5

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

    if 5.3206985442429606e+193 < alpha

    1. Initial program 64.0

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 39.2

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

    4. Simplified39.2

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, 1 \cdot 1, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\right)\right)}{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) + \left(1 \cdot 1 - \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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))