Average Error: 3.8 → 2.5
Time: 26.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.056793302986320848262884903425314505228 \cdot 10^{175}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{{\alpha}^{2}} + \left(1 - \frac{1}{\alpha}\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \end{array}\]
\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}\;\alpha \le 5.056793302986320848262884903425314505228 \cdot 10^{175}:\\
\;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r78180 = alpha;
        double r78181 = beta;
        double r78182 = r78180 + r78181;
        double r78183 = r78181 * r78180;
        double r78184 = r78182 + r78183;
        double r78185 = 1.0;
        double r78186 = r78184 + r78185;
        double r78187 = 2.0;
        double r78188 = r78187 * r78185;
        double r78189 = r78182 + r78188;
        double r78190 = r78186 / r78189;
        double r78191 = r78190 / r78189;
        double r78192 = r78189 + r78185;
        double r78193 = r78191 / r78192;
        return r78193;
}
↓
double f(double alpha, double beta) {
        double r78194 = alpha;
        double r78195 = 5.056793302986321e+175;
        bool r78196 = r78194 <= r78195;
        double r78197 = 1.0;
        double r78198 = beta;
        double r78199 = r78194 + r78198;
        double r78200 = fma(r78194, r78198, r78199);
        double r78201 = r78197 + r78200;
        double r78202 = 2.0;
        double r78203 = fma(r78197, r78202, r78199);
        double r78204 = r78201 / r78203;
        double r78205 = r78204 / r78203;
        double r78206 = r78197 * r78202;
        double r78207 = r78199 + r78197;
        double r78208 = r78206 + r78207;
        double r78209 = r78205 / r78208;
        double r78210 = 2.0;
        double r78211 = pow(r78194, r78210);
        double r78212 = r78202 / r78211;
        double r78213 = 1.0;
        double r78214 = r78197 / r78194;
        double r78215 = r78213 - r78214;
        double r78216 = r78212 + r78215;
        double r78217 = r78216 / r78203;
        double r78218 = r78217 / r78208;
        double r78219 = r78196 ? r78209 : r78218;
        return r78219;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.056793302986321e+175

    1. Initial program 1.7

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

    2. Simplified1.7

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

    4. Applied fma-udef1.7

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

    5. Applied associate-+l+1.7

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

    if 5.056793302986321e+175 < alpha

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied fma-udef16.3

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

    5. Applied associate-+l+16.3

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

    6. Taylor expanded around inf 7.4

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

    7. Simplified7.4

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.056793302986320848262884903425314505228 \cdot 10^{175}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{{\alpha}^{2}} + \left(1 - \frac{1}{\alpha}\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))