Average Error: 3.6 → 2.8
Time: 25.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, 0.25, \mathsf{fma}\left(0.25, \beta, 0.5\right)\right)}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\alpha, 0.25, \mathsf{fma}\left(0.25, \beta, 0.5\right)\right)}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\

\end{array}
double f(double alpha, double beta) {
        double r3323226 = alpha;
        double r3323227 = beta;
        double r3323228 = r3323226 + r3323227;
        double r3323229 = r3323227 * r3323226;
        double r3323230 = r3323228 + r3323229;
        double r3323231 = 1.0;
        double r3323232 = r3323230 + r3323231;
        double r3323233 = 2.0;
        double r3323234 = 1.0;
        double r3323235 = r3323233 * r3323234;
        double r3323236 = r3323228 + r3323235;
        double r3323237 = r3323232 / r3323236;
        double r3323238 = r3323237 / r3323236;
        double r3323239 = r3323236 + r3323231;
        double r3323240 = r3323238 / r3323239;
        return r3323240;
}
↓
double f(double alpha, double beta) {
        double r3323241 = beta;
        double r3323242 = 1.3353325288374214e+154;
        bool r3323243 = r3323241 <= r3323242;
        double r3323244 = 1.0;
        double r3323245 = alpha;
        double r3323246 = r3323245 * r3323241;
        double r3323247 = r3323245 + r3323241;
        double r3323248 = r3323246 + r3323247;
        double r3323249 = r3323244 + r3323248;
        double r3323250 = 2.0;
        double r3323251 = r3323247 + r3323250;
        double r3323252 = r3323249 / r3323251;
        double r3323253 = r3323244 + r3323251;
        double r3323254 = r3323253 * r3323247;
        double r3323255 = r3323253 * r3323250;
        double r3323256 = r3323254 + r3323255;
        double r3323257 = r3323252 / r3323256;
        double r3323258 = 0.25;
        double r3323259 = 0.5;
        double r3323260 = fma(r3323258, r3323241, r3323259);
        double r3323261 = fma(r3323245, r3323258, r3323260);
        double r3323262 = r3323261 / r3323256;
        double r3323263 = r3323243 ? r3323257 : r3323262;
        return r3323263;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.3353325288374214e+154

    1. Initial program 1.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied associate-/l/1.6

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}\]
    4. Using strategy rm

    5. Applied distribute-rgt-in1.6

      \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) + \left(2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}}\]

    if 1.3353325288374214e+154 < beta

    1. Initial program 16.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied associate-/l/18.7

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}\]
    4. Using strategy rm

    5. Applied distribute-rgt-in18.7

      \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\alpha + \beta\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) + \left(2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}}\]

    6. Taylor expanded around 0 8.9

      \[\leadsto \frac{\color{blue}{0.25 \cdot \alpha + \left(0.25 \cdot \beta + 0.5\right)}}{\left(\alpha + \beta\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) + \left(2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]

    7. Simplified8.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha, 0.25, \mathsf{fma}\left(0.25, \beta, 0.5\right)\right)}}{\left(\alpha + \beta\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) + \left(2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, 0.25, \mathsf{fma}\left(0.25, \beta, 0.5\right)\right)}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\alpha + \beta\right) + \left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))