Average Error: 3.6 → 1.1
Time: 50.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.3403637264423054 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}{\frac{2 + \left(\alpha + \beta\right)}{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)\right) \cdot \left(\left(1.0 + 2\right) + \left(\alpha + \beta\right)\right)}\\ \end{array}\]
\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 2.3403637264423054 \cdot 10^{+157}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}{\frac{2 + \left(\alpha + \beta\right)}{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)\right) \cdot \left(\left(1.0 + 2\right) + \left(\alpha + \beta\right)\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r4959220 = alpha;
        double r4959221 = beta;
        double r4959222 = r4959220 + r4959221;
        double r4959223 = r4959221 * r4959220;
        double r4959224 = r4959222 + r4959223;
        double r4959225 = 1.0;
        double r4959226 = r4959224 + r4959225;
        double r4959227 = 2.0;
        double r4959228 = 1.0;
        double r4959229 = r4959227 * r4959228;
        double r4959230 = r4959222 + r4959229;
        double r4959231 = r4959226 / r4959230;
        double r4959232 = r4959231 / r4959230;
        double r4959233 = r4959230 + r4959225;
        double r4959234 = r4959232 / r4959233;
        return r4959234;
}
↓
double f(double alpha, double beta) {
        double r4959235 = beta;
        double r4959236 = 2.3403637264423054e+157;
        bool r4959237 = r4959235 <= r4959236;
        double r4959238 = 1.0;
        double r4959239 = alpha;
        double r4959240 = r4959239 * r4959235;
        double r4959241 = r4959239 + r4959235;
        double r4959242 = r4959240 + r4959241;
        double r4959243 = r4959238 + r4959242;
        double r4959244 = sqrt(r4959243);
        double r4959245 = 2.0;
        double r4959246 = r4959245 + r4959241;
        double r4959247 = sqrt(r4959246);
        double r4959248 = r4959244 / r4959247;
        double r4959249 = r4959246 / r4959248;
        double r4959250 = r4959248 / r4959249;
        double r4959251 = r4959246 + r4959238;
        double r4959252 = r4959250 / r4959251;
        double r4959253 = 1.0;
        double r4959254 = r4959235 / r4959239;
        double r4959255 = r4959239 / r4959235;
        double r4959256 = r4959255 + r4959245;
        double r4959257 = r4959254 + r4959256;
        double r4959258 = r4959238 + r4959245;
        double r4959259 = r4959258 + r4959241;
        double r4959260 = r4959257 * r4959259;
        double r4959261 = r4959253 / r4959260;
        double r4959262 = r4959237 ? r4959252 : r4959261;
        return r4959262;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.3403637264423054e+157

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.7

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

    4. Applied add-sqr-sqrt1.7

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

    5. Applied times-frac1.7

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

    6. Applied associate-/l*1.2

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

    if 2.3403637264423054e+157 < beta

    1. Initial program 16.3

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

    3. Applied clear-num16.3

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

    4. Applied associate-/l/16.3

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

    5. Applied associate-/l/16.7

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

    6. Simplified16.7

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

    7. Taylor expanded around inf 0.9

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.3403637264423054 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}{\frac{2 + \left(\alpha + \beta\right)}{\frac{\sqrt{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{2 + \left(\alpha + \beta\right)}}}}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)\right) \cdot \left(\left(1.0 + 2\right) + \left(\alpha + \beta\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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)))