Average Error: 16.0 → 3.3
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.99999999999924083:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\mathsf{fma}\left(1, 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.99999999999924083:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r107345 = beta;
        double r107346 = alpha;
        double r107347 = r107345 - r107346;
        double r107348 = r107346 + r107345;
        double r107349 = 2.0;
        double r107350 = r107348 + r107349;
        double r107351 = r107347 / r107350;
        double r107352 = 1.0;
        double r107353 = r107351 + r107352;
        double r107354 = r107353 / r107349;
        return r107354;
}
↓
double f(double alpha, double beta) {
        double r107355 = beta;
        double r107356 = alpha;
        double r107357 = r107355 - r107356;
        double r107358 = r107356 + r107355;
        double r107359 = 2.0;
        double r107360 = r107358 + r107359;
        double r107361 = r107357 / r107360;
        double r107362 = -0.9999999999992408;
        bool r107363 = r107361 <= r107362;
        double r107364 = cbrt(r107355);
        double r107365 = r107364 * r107364;
        double r107366 = cbrt(r107360);
        double r107367 = r107366 * r107366;
        double r107368 = r107365 / r107367;
        double r107369 = r107364 / r107366;
        double r107370 = r107368 * r107369;
        double r107371 = 4.0;
        double r107372 = 1.0;
        double r107373 = 2.0;
        double r107374 = pow(r107356, r107373);
        double r107375 = r107372 / r107374;
        double r107376 = r107372 / r107356;
        double r107377 = 8.0;
        double r107378 = 3.0;
        double r107379 = pow(r107356, r107378);
        double r107380 = r107372 / r107379;
        double r107381 = r107377 * r107380;
        double r107382 = fma(r107359, r107376, r107381);
        double r107383 = -r107382;
        double r107384 = fma(r107371, r107375, r107383);
        double r107385 = r107370 - r107384;
        double r107386 = r107385 / r107359;
        double r107387 = pow(r107361, r107378);
        double r107388 = 1.0;
        double r107389 = pow(r107388, r107378);
        double r107390 = r107387 + r107389;
        double r107391 = r107388 - r107361;
        double r107392 = r107361 * r107361;
        double r107393 = fma(r107388, r107391, r107392);
        double r107394 = r107390 / r107393;
        double r107395 = r107394 / r107359;
        double r107396 = r107363 ? r107386 : r107395;
        return r107396;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999992408

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.4

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

    6. Applied add-cube-cbrt58.4

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

    7. Applied add-cube-cbrt58.4

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

    8. Applied times-frac58.4

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

    9. Taylor expanded around inf 11.6

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

    10. Simplified11.6

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

    if -0.9999999999992408 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.3

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

    3. Applied flip3-+0.3

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

    4. Simplified0.3

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.99999999999924083:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\mathsf{fma}\left(1, 1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))