Average Error: 16.4 → 6.0
Time: 6.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6769387663364106240:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6769387663364106240:\\
\;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r109105 = beta;
        double r109106 = alpha;
        double r109107 = r109105 - r109106;
        double r109108 = r109106 + r109105;
        double r109109 = 2.0;
        double r109110 = r109108 + r109109;
        double r109111 = r109107 / r109110;
        double r109112 = 1.0;
        double r109113 = r109111 + r109112;
        double r109114 = r109113 / r109109;
        return r109114;
}
↓
double f(double alpha, double beta) {
        double r109115 = alpha;
        double r109116 = 6.769387663364106e+18;
        bool r109117 = r109115 <= r109116;
        double r109118 = 1.0;
        double r109119 = sqrt(r109118);
        double r109120 = beta;
        double r109121 = r109115 + r109120;
        double r109122 = 2.0;
        double r109123 = r109121 + r109122;
        double r109124 = r109119 / r109123;
        double r109125 = r109118 / r109120;
        double r109126 = r109119 / r109125;
        double r109127 = r109115 / r109123;
        double r109128 = 1.0;
        double r109129 = r109127 - r109128;
        double r109130 = -r109129;
        double r109131 = fma(r109124, r109126, r109130);
        double r109132 = log(r109131);
        double r109133 = exp(r109132);
        double r109134 = r109133 / r109122;
        double r109135 = 4.0;
        double r109136 = 2.0;
        double r109137 = pow(r109115, r109136);
        double r109138 = r109118 / r109137;
        double r109139 = r109118 / r109115;
        double r109140 = 8.0;
        double r109141 = 3.0;
        double r109142 = pow(r109115, r109141);
        double r109143 = r109118 / r109142;
        double r109144 = r109140 * r109143;
        double r109145 = fma(r109122, r109139, r109144);
        double r109146 = -r109145;
        double r109147 = fma(r109135, r109138, r109146);
        double r109148 = -r109147;
        double r109149 = fma(r109124, r109126, r109148);
        double r109150 = r109149 / r109122;
        double r109151 = r109117 ? r109134 : r109150;
        return r109151;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.769387663364106e+18

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    6. Applied clear-num0.7

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    7. Using strategy rm

    8. Applied div-inv0.7

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

    9. Applied add-sqr-sqrt0.7

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

    10. Applied times-frac0.7

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

    11. Applied fma-neg0.7

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

    13. Applied add-exp-log0.7

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

    if 6.769387663364106e+18 < alpha

    1. Initial program 51.0

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

    3. Applied div-sub50.9

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

    4. Applied associate-+l-49.3

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

    6. Applied clear-num49.4

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    7. Using strategy rm

    8. Applied div-inv49.4

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

    9. Applied add-sqr-sqrt49.4

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

    10. Applied times-frac49.3

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

    11. Applied fma-neg49.3

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

    12. Taylor expanded around inf 17.6

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right)}{2}\]

    13. Simplified17.6

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right)}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6769387663364106240:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{1}}{\left(\alpha + \beta\right) + 2}, \frac{\sqrt{1}}{\frac{1}{\beta}}, -\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))