Average Error: 3.8 → 2.8
Time: 24.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.795069092854939666192787143111574910834 \cdot 10^{91}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\right)\right) \cdot \frac{1}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\beta}\right) + \frac{2}{\beta \cdot \beta}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 2.795069092854939666192787143111574910834 \cdot 10^{91}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\right)\right) \cdot \frac{1}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(1 - \frac{1}{\beta}\right) + \frac{2}{\beta \cdot \beta}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r3842634 = alpha;
        double r3842635 = beta;
        double r3842636 = r3842634 + r3842635;
        double r3842637 = r3842635 * r3842634;
        double r3842638 = r3842636 + r3842637;
        double r3842639 = 1.0;
        double r3842640 = r3842638 + r3842639;
        double r3842641 = 2.0;
        double r3842642 = r3842641 * r3842639;
        double r3842643 = r3842636 + r3842642;
        double r3842644 = r3842640 / r3842643;
        double r3842645 = r3842644 / r3842643;
        double r3842646 = r3842643 + r3842639;
        double r3842647 = r3842645 / r3842646;
        return r3842647;
}
↓
double f(double alpha, double beta) {
        double r3842648 = beta;
        double r3842649 = 2.7950690928549397e+91;
        bool r3842650 = r3842648 <= r3842649;
        double r3842651 = 1.0;
        double r3842652 = alpha;
        double r3842653 = r3842648 + r3842652;
        double r3842654 = fma(r3842648, r3842652, r3842653);
        double r3842655 = r3842651 + r3842654;
        double r3842656 = 2.0;
        double r3842657 = fma(r3842656, r3842651, r3842653);
        double r3842658 = r3842655 / r3842657;
        double r3842659 = r3842658 / r3842657;
        double r3842660 = expm1(r3842659);
        double r3842661 = log1p(r3842660);
        double r3842662 = 1.0;
        double r3842663 = r3842651 + r3842657;
        double r3842664 = r3842662 / r3842663;
        double r3842665 = r3842661 * r3842664;
        double r3842666 = r3842651 / r3842648;
        double r3842667 = r3842662 - r3842666;
        double r3842668 = r3842648 * r3842648;
        double r3842669 = r3842656 / r3842668;
        double r3842670 = r3842667 + r3842669;
        double r3842671 = r3842670 / r3842657;
        double r3842672 = r3842671 / r3842663;
        double r3842673 = r3842650 ? r3842665 : r3842672;
        return r3842673;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.7950690928549397e+91

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied log1p-expm1-u0.4

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\right)\right)}}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\]
    5. Using strategy rm

    6. Applied div-inv0.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\right)\right) \cdot \frac{1}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}}\]

    if 2.7950690928549397e+91 < beta

    1. Initial program 14.4

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

    2. Simplified14.4

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

    3. Taylor expanded around inf 10.1

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

    4. Simplified10.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{\beta \cdot \beta} + \left(1 - \frac{1}{\beta}\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.795069092854939666192787143111574910834 \cdot 10^{91}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\right)\right) \cdot \frac{1}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\beta}\right) + \frac{2}{\beta \cdot \beta}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{1 + \mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))