Average Error: 16.5 → 6.4
Time: 5.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 121320078.19471414387226104736328125:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\frac{\frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\sqrt{\left(\alpha + \beta\right) + 2}} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 121320078.19471414387226104736328125:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\frac{\frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\sqrt{\left(\alpha + \beta\right) + 2}} + 1}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r98101 = beta;
        double r98102 = alpha;
        double r98103 = r98101 - r98102;
        double r98104 = r98102 + r98101;
        double r98105 = 2.0;
        double r98106 = r98104 + r98105;
        double r98107 = r98103 / r98106;
        double r98108 = 1.0;
        double r98109 = r98107 + r98108;
        double r98110 = r98109 / r98105;
        return r98110;
}
↓
double f(double alpha, double beta) {
        double r98111 = alpha;
        double r98112 = 121320078.19471414;
        bool r98113 = r98111 <= r98112;
        double r98114 = beta;
        double r98115 = r98111 + r98114;
        double r98116 = 2.0;
        double r98117 = r98115 + r98116;
        double r98118 = r98114 / r98117;
        double r98119 = 1.0;
        double r98120 = r98119 * r98119;
        double r98121 = -r98120;
        double r98122 = r98111 / r98117;
        double r98123 = r98122 * r98111;
        double r98124 = r98123 / r98117;
        double r98125 = r98121 + r98124;
        double r98126 = sqrt(r98117);
        double r98127 = r98111 / r98126;
        double r98128 = r98127 / r98126;
        double r98129 = r98128 + r98119;
        double r98130 = r98125 / r98129;
        double r98131 = r98118 - r98130;
        double r98132 = r98131 / r98116;
        double r98133 = 4.0;
        double r98134 = r98133 / r98111;
        double r98135 = r98134 / r98111;
        double r98136 = r98116 / r98111;
        double r98137 = 8.0;
        double r98138 = -r98137;
        double r98139 = 3.0;
        double r98140 = pow(r98111, r98139);
        double r98141 = r98138 / r98140;
        double r98142 = r98136 - r98141;
        double r98143 = r98135 - r98142;
        double r98144 = r98118 - r98143;
        double r98145 = r98144 / r98116;
        double r98146 = r98113 ? r98132 : r98145;
        return r98146;
}

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 alpha < 121320078.19471414

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied associate-/r*0.1

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

    9. Applied flip--0.1

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

    10. Simplified0.1

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

    if 121320078.19471414 < alpha

    1. Initial program 49.3

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

    3. Applied div-sub49.2

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

    4. Applied associate-+l-47.7

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

    5. Taylor expanded around inf 18.9

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

    6. Simplified18.9

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 121320078.19471414387226104736328125:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\frac{\frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\sqrt{\left(\alpha + \beta\right) + 2}} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))