Average Error: 15.9 → 6.0
Time: 21.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2} \cdot \beta - \mathsf{fma}\left(\sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\
\;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2} \cdot \beta - \mathsf{fma}\left(\sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, -1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3972567 = beta;
        double r3972568 = alpha;
        double r3972569 = r3972567 - r3972568;
        double r3972570 = r3972568 + r3972567;
        double r3972571 = 2.0;
        double r3972572 = r3972570 + r3972571;
        double r3972573 = r3972569 / r3972572;
        double r3972574 = 1.0;
        double r3972575 = r3972573 + r3972574;
        double r3972576 = r3972575 / r3972571;
        return r3972576;
}
↓
double f(double alpha, double beta) {
        double r3972577 = alpha;
        double r3972578 = 644102.6150741008;
        bool r3972579 = r3972577 <= r3972578;
        double r3972580 = 1.0;
        double r3972581 = beta;
        double r3972582 = r3972581 + r3972577;
        double r3972583 = 2.0;
        double r3972584 = r3972582 + r3972583;
        double r3972585 = r3972580 / r3972584;
        double r3972586 = r3972585 * r3972581;
        double r3972587 = r3972577 / r3972584;
        double r3972588 = cbrt(r3972587);
        double r3972589 = r3972588 * r3972588;
        double r3972590 = 1.0;
        double r3972591 = -r3972590;
        double r3972592 = fma(r3972589, r3972588, r3972591);
        double r3972593 = r3972586 - r3972592;
        double r3972594 = r3972593 / r3972583;
        double r3972595 = r3972581 / r3972584;
        double r3972596 = 4.0;
        double r3972597 = r3972577 * r3972577;
        double r3972598 = r3972596 / r3972597;
        double r3972599 = r3972583 / r3972577;
        double r3972600 = r3972598 - r3972599;
        double r3972601 = 8.0;
        double r3972602 = r3972577 * r3972597;
        double r3972603 = r3972601 / r3972602;
        double r3972604 = r3972600 - r3972603;
        double r3972605 = r3972595 - r3972604;
        double r3972606 = r3972605 / r3972583;
        double r3972607 = r3972579 ? r3972594 : r3972606;
        return r3972607;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 644102.6150741008

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied div-inv0.1

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied fma-neg0.1

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

    if 644102.6150741008 < alpha

    1. Initial program 49.6

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

    3. Applied div-sub49.5

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

    4. Applied associate-+l-47.9

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

    5. Taylor expanded around inf 18.6

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

    6. Simplified18.6

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\ \;\;\;\;\frac{\frac{1}{\left(\beta + \alpha\right) + 2} \cdot \beta - \mathsf{fma}\left(\sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, \sqrt[3]{\frac{\alpha}{\left(\beta + \alpha\right) + 2}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))