Average Error: 16.5 → 6.0
Time: 4.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 410714263654.793396:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} + \sqrt{{\left(1 \cdot 1\right)}^{3}}\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} - \sqrt{{\left(1 \cdot 1\right)}^{3}}\right)}{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{4} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 410714263654.793396:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} + \sqrt{{\left(1 \cdot 1\right)}^{3}}\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} - \sqrt{{\left(1 \cdot 1\right)}^{3}}\right)}{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{4} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r87203 = beta;
        double r87204 = alpha;
        double r87205 = r87203 - r87204;
        double r87206 = r87204 + r87203;
        double r87207 = 2.0;
        double r87208 = r87206 + r87207;
        double r87209 = r87205 / r87208;
        double r87210 = 1.0;
        double r87211 = r87209 + r87210;
        double r87212 = r87211 / r87207;
        return r87212;
}
↓
double f(double alpha, double beta) {
        double r87213 = alpha;
        double r87214 = 410714263654.7934;
        bool r87215 = r87213 <= r87214;
        double r87216 = beta;
        double r87217 = r87213 + r87216;
        double r87218 = 2.0;
        double r87219 = r87217 + r87218;
        double r87220 = r87216 / r87219;
        double r87221 = r87213 / r87219;
        double r87222 = r87221 * r87221;
        double r87223 = 1.5;
        double r87224 = pow(r87222, r87223);
        double r87225 = 1.0;
        double r87226 = r87225 * r87225;
        double r87227 = 3.0;
        double r87228 = pow(r87226, r87227);
        double r87229 = sqrt(r87228);
        double r87230 = r87224 + r87229;
        double r87231 = r87224 - r87229;
        double r87232 = r87230 * r87231;
        double r87233 = 4.0;
        double r87234 = pow(r87221, r87233);
        double r87235 = r87222 * r87226;
        double r87236 = r87234 + r87235;
        double r87237 = pow(r87225, r87233);
        double r87238 = r87236 + r87237;
        double r87239 = r87221 + r87225;
        double r87240 = r87238 * r87239;
        double r87241 = r87232 / r87240;
        double r87242 = r87220 - r87241;
        double r87243 = r87242 / r87218;
        double r87244 = 4.0;
        double r87245 = r87244 / r87213;
        double r87246 = r87245 / r87213;
        double r87247 = 8.0;
        double r87248 = -r87247;
        double r87249 = pow(r87213, r87227);
        double r87250 = r87248 / r87249;
        double r87251 = r87246 + r87250;
        double r87252 = -r87218;
        double r87253 = r87252 / r87213;
        double r87254 = r87251 + r87253;
        double r87255 = r87220 - r87254;
        double r87256 = r87255 / r87218;
        double r87257 = r87215 ? r87243 : r87256;
        return r87257;
}

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 < 410714263654.7934

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied flip--0.2

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

    8. Applied flip3--0.2

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

    9. Applied associate-/l/0.2

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

    10. Simplified0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\color{blue}{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{4} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}}{2}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt0.2

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

    13. Applied sqr-pow0.2

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

    14. Applied difference-of-squares0.2

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

    15. Simplified0.2

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

    16. Simplified0.2

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

    if 410714263654.7934 < alpha

    1. Initial program 50.2

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

    3. Applied div-sub50.2

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

    4. Applied associate-+l-48.6

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

    5. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 410714263654.793396:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} + \sqrt{{\left(1 \cdot 1\right)}^{3}}\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{\frac{3}{2}} - \sqrt{{\left(1 \cdot 1\right)}^{3}}\right)}{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{4} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))