Average Error: 3.7 → 2.4
Time: 1.5m
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 r4056927 = alpha;
        double r4056928 = beta;
        double r4056929 = r4056927 + r4056928;
        double r4056930 = r4056928 * r4056927;
        double r4056931 = r4056929 + r4056930;
        double r4056932 = 1.0;
        double r4056933 = r4056931 + r4056932;
        double r4056934 = 2.0;
        double r4056935 = r4056934 * r4056932;
        double r4056936 = r4056929 + r4056935;
        double r4056937 = r4056933 / r4056936;
        double r4056938 = r4056937 / r4056936;
        double r4056939 = r4056936 + r4056932;
        double r4056940 = r4056938 / r4056939;
        return r4056940;
}
↓
double f(double alpha, double beta) {
        double r4056941 = beta;
        double r4056942 = 1.8833927498580965e+214;
        bool r4056943 = r4056941 <= r4056942;
        double r4056944 = alpha;
        double r4056945 = 1.0;
        double r4056946 = r4056944 + r4056941;
        double r4056947 = r4056945 + r4056946;
        double r4056948 = fma(r4056944, r4056941, r4056947);
        double r4056949 = 2.0;
        double r4056950 = fma(r4056945, r4056949, r4056946);
        double r4056951 = r4056948 / r4056950;
        double r4056952 = 1.0;
        double r4056953 = r4056952 / r4056950;
        double r4056954 = r4056951 * r4056953;
        double r4056955 = r4056950 + r4056945;
        double r4056956 = r4056954 / r4056955;
        double r4056957 = 0.5;
        double r4056958 = sqrt(r4056957);
        double r4056959 = 0.75;
        double r4056960 = r4056944 * r4056959;
        double r4056961 = r4056945 + r4056960;
        double r4056962 = r4056958 * r4056961;
        double r4056963 = 0.125;
        double r4056964 = r4056958 / r4056941;
        double r4056965 = r4056963 / r4056964;
        double r4056966 = r4056962 - r4056965;
        double r4056967 = fma(r4056958, r4056941, r4056966);
        double r4056968 = sqrt(r4056950);
        double r4056969 = r4056967 / r4056968;
        double r4056970 = r4056969 * r4056953;
        double r4056971 = r4056970 / r4056955;
        double r4056972 = r4056943 ? r4056956 : r4056971;
        return r4056972;
}

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)))