Average Error: 3.7 → 2.4
Time: 54.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.883392749858096523170277083069548182618 \cdot 10^{214}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, 1 + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{0.5}, \beta, \sqrt{0.5} \cdot \left(1 + \alpha \cdot 0.75\right) - \frac{0.125}{\frac{\sqrt{0.5}}{\beta}}\right)}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]
\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 1.883392749858096523170277083069548182618 \cdot 10^{214}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, 1 + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{0.5}, \beta, \sqrt{0.5} \cdot \left(1 + \alpha \cdot 0.75\right) - \frac{0.125}{\frac{\sqrt{0.5}}{\beta}}\right)}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r4087236 = alpha;
        double r4087237 = beta;
        double r4087238 = r4087236 + r4087237;
        double r4087239 = r4087237 * r4087236;
        double r4087240 = r4087238 + r4087239;
        double r4087241 = 1.0;
        double r4087242 = r4087240 + r4087241;
        double r4087243 = 2.0;
        double r4087244 = r4087243 * r4087241;
        double r4087245 = r4087238 + r4087244;
        double r4087246 = r4087242 / r4087245;
        double r4087247 = r4087246 / r4087245;
        double r4087248 = r4087245 + r4087241;
        double r4087249 = r4087247 / r4087248;
        return r4087249;
}
↓
double f(double alpha, double beta) {
        double r4087250 = beta;
        double r4087251 = 1.8833927498580965e+214;
        bool r4087252 = r4087250 <= r4087251;
        double r4087253 = alpha;
        double r4087254 = 1.0;
        double r4087255 = r4087253 + r4087250;
        double r4087256 = r4087254 + r4087255;
        double r4087257 = fma(r4087253, r4087250, r4087256);
        double r4087258 = 2.0;
        double r4087259 = fma(r4087254, r4087258, r4087255);
        double r4087260 = r4087257 / r4087259;
        double r4087261 = 1.0;
        double r4087262 = r4087261 / r4087259;
        double r4087263 = r4087260 * r4087262;
        double r4087264 = r4087259 + r4087254;
        double r4087265 = r4087263 / r4087264;
        double r4087266 = 0.5;
        double r4087267 = sqrt(r4087266);
        double r4087268 = 0.75;
        double r4087269 = r4087253 * r4087268;
        double r4087270 = r4087254 + r4087269;
        double r4087271 = r4087267 * r4087270;
        double r4087272 = 0.125;
        double r4087273 = r4087267 / r4087250;
        double r4087274 = r4087272 / r4087273;
        double r4087275 = r4087271 - r4087274;
        double r4087276 = fma(r4087267, r4087250, r4087275);
        double r4087277 = sqrt(r4087259);
        double r4087278 = r4087276 / r4087277;
        double r4087279 = r4087278 * r4087262;
        double r4087280 = r4087279 / r4087264;
        double r4087281 = r4087252 ? r4087265 : r4087280;
        return r4087281;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.8833927498580965e+214

    1. Initial program 2.1

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

    2. Simplified2.1

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

    4. Applied div-inv2.1

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

    if 1.8833927498580965e+214 < beta

    1. Initial program 17.6

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

    2. Simplified17.6

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

    4. Applied div-inv17.6

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

    6. Applied add-sqr-sqrt17.6

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

    7. Applied associate-/r*17.6

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

    8. Taylor expanded around 0 5.0

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]

    9. Simplified4.9

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \beta, \sqrt{0.5} \cdot \left(1 + 0.75 \cdot \alpha\right) - \frac{0.125}{\frac{\sqrt{0.5}}{\beta}}\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.883392749858096523170277083069548182618 \cdot 10^{214}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, 1 + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{0.5}, \beta, \sqrt{0.5} \cdot \left(1 + \alpha \cdot 0.75\right) - \frac{0.125}{\frac{\sqrt{0.5}}{\beta}}\right)}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))