Average Error: 3.6 → 2.7
Time: 15.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 7.494253796401252134956512833080212050133 \cdot 10^{146}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\beta \le 7.494253796401252134956512833080212050133 \cdot 10^{146}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\

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

\end{array}
double f(double alpha, double beta) {
        double r129963 = alpha;
        double r129964 = beta;
        double r129965 = r129963 + r129964;
        double r129966 = r129964 * r129963;
        double r129967 = r129965 + r129966;
        double r129968 = 1.0;
        double r129969 = r129967 + r129968;
        double r129970 = 2.0;
        double r129971 = r129970 * r129968;
        double r129972 = r129965 + r129971;
        double r129973 = r129969 / r129972;
        double r129974 = r129973 / r129972;
        double r129975 = r129972 + r129968;
        double r129976 = r129974 / r129975;
        return r129976;
}
↓
double f(double alpha, double beta) {
        double r129977 = beta;
        double r129978 = 7.494253796401252e+146;
        bool r129979 = r129977 <= r129978;
        double r129980 = 1.0;
        double r129981 = alpha;
        double r129982 = r129981 + r129977;
        double r129983 = fma(r129981, r129977, r129982);
        double r129984 = r129980 + r129983;
        double r129985 = sqrt(r129984);
        double r129986 = 2.0;
        double r129987 = fma(r129980, r129986, r129982);
        double r129988 = sqrt(r129987);
        double r129989 = r129985 / r129988;
        double r129990 = r129989 / r129988;
        double r129991 = r129987 + r129980;
        double r129992 = r129985 / r129987;
        double r129993 = r129991 / r129992;
        double r129994 = r129990 / r129993;
        double r129995 = 0.0;
        double r129996 = r129979 ? r129994 : r129995;
        return r129996;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 7.494253796401252e+146

    1. Initial program 1.1

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

    2. Simplified1.1

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

    4. Applied add-sqr-sqrt1.7

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

    5. Applied add-sqr-sqrt2.1

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

    6. Applied add-sqr-sqrt2.0

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

    7. Applied times-frac2.0

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

    8. Applied times-frac1.7

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

    9. Applied associate-/l*1.7

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

    10. Simplified1.2

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

    if 7.494253796401252e+146 < beta

    1. Initial program 15.2

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

    2. Simplified15.2

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

    4. Applied expm1-log1p-u15.2

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\right)\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]

    5. Taylor expanded around inf 10.1

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.494253796401252134956512833080212050133 \cdot 10^{146}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))