Average Error: 3.6 → 2.2
Time: 2.4m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.247457934878237765470825934048239254403 \cdot 10^{194}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 1.247457934878237765470825934048239254403 \cdot 10^{194}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r14759336 = alpha;
        double r14759337 = beta;
        double r14759338 = r14759336 + r14759337;
        double r14759339 = r14759337 * r14759336;
        double r14759340 = r14759338 + r14759339;
        double r14759341 = 1.0;
        double r14759342 = r14759340 + r14759341;
        double r14759343 = 2.0;
        double r14759344 = r14759343 * r14759341;
        double r14759345 = r14759338 + r14759344;
        double r14759346 = r14759342 / r14759345;
        double r14759347 = r14759346 / r14759345;
        double r14759348 = r14759345 + r14759341;
        double r14759349 = r14759347 / r14759348;
        return r14759349;
}
↓
double f(double alpha, double beta) {
        double r14759350 = alpha;
        double r14759351 = 1.2474579348782378e+194;
        bool r14759352 = r14759350 <= r14759351;
        double r14759353 = 1.0;
        double r14759354 = beta;
        double r14759355 = r14759350 + r14759354;
        double r14759356 = fma(r14759354, r14759350, r14759355);
        double r14759357 = r14759353 + r14759356;
        double r14759358 = sqrt(r14759357);
        double r14759359 = 2.0;
        double r14759360 = fma(r14759359, r14759353, r14759355);
        double r14759361 = sqrt(r14759360);
        double r14759362 = r14759358 / r14759361;
        double r14759363 = r14759360 / r14759362;
        double r14759364 = r14759362 / r14759363;
        double r14759365 = r14759353 + r14759360;
        double r14759366 = r14759364 / r14759365;
        double r14759367 = 0.0;
        double r14759368 = r14759352 ? r14759366 : r14759367;
        return r14759368;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.2474579348782378e+194

    1. Initial program 1.8

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

    2. Simplified1.8

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

    4. Applied add-sqr-sqrt2.4

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

    5. Applied add-sqr-sqrt2.3

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

    6. Applied times-frac2.3

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

    7. Applied associate-/l*1.9

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

    if 1.2474579348782378e+194 < alpha

    1. Initial program 16.5

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

    2. Simplified16.5

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

    3. Taylor expanded around inf 4.3

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.247457934878237765470825934048239254403 \cdot 10^{194}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\frac{\sqrt{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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)))