Average Error: 23.9 → 11.8
Time: 18.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.434843859125862 \cdot 10^{+184}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right)\right)}\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)\right)}}{2.0}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 8.434843859125862 \cdot 10^{+184}:\\
\;\;\;\;\frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right)\right)}\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)\right)}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r3708639 = alpha;
        double r3708640 = beta;
        double r3708641 = r3708639 + r3708640;
        double r3708642 = r3708640 - r3708639;
        double r3708643 = r3708641 * r3708642;
        double r3708644 = 2.0;
        double r3708645 = i;
        double r3708646 = r3708644 * r3708645;
        double r3708647 = r3708641 + r3708646;
        double r3708648 = r3708643 / r3708647;
        double r3708649 = 2.0;
        double r3708650 = r3708647 + r3708649;
        double r3708651 = r3708648 / r3708650;
        double r3708652 = 1.0;
        double r3708653 = r3708651 + r3708652;
        double r3708654 = r3708653 / r3708649;
        return r3708654;
}
↓
double f(double alpha, double beta, double i) {
        double r3708655 = alpha;
        double r3708656 = 8.434843859125862e+184;
        bool r3708657 = r3708655 <= r3708656;
        double r3708658 = 1.0;
        double r3708659 = i;
        double r3708660 = 2.0;
        double r3708661 = beta;
        double r3708662 = r3708655 + r3708661;
        double r3708663 = fma(r3708659, r3708660, r3708662);
        double r3708664 = r3708663 / r3708662;
        double r3708665 = r3708658 / r3708664;
        double r3708666 = r3708661 - r3708655;
        double r3708667 = 2.0;
        double r3708668 = r3708667 + r3708663;
        double r3708669 = r3708666 / r3708668;
        double r3708670 = 1.0;
        double r3708671 = fma(r3708665, r3708669, r3708670);
        double r3708672 = r3708671 * r3708671;
        double r3708673 = r3708671 * r3708672;
        double r3708674 = cbrt(r3708673);
        double r3708675 = log(r3708674);
        double r3708676 = exp(r3708675);
        double r3708677 = r3708676 / r3708667;
        double r3708678 = 8.0;
        double r3708679 = r3708678 / r3708655;
        double r3708680 = r3708655 * r3708655;
        double r3708681 = r3708679 / r3708680;
        double r3708682 = r3708667 / r3708655;
        double r3708683 = 4.0;
        double r3708684 = r3708683 / r3708680;
        double r3708685 = r3708682 - r3708684;
        double r3708686 = r3708681 + r3708685;
        double r3708687 = log(r3708686);
        double r3708688 = exp(r3708687);
        double r3708689 = r3708688 / r3708667;
        double r3708690 = r3708657 ? r3708677 : r3708689;
        return r3708690;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 8.434843859125862e+184

    1. Initial program 18.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Simplified7.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}, \frac{\beta - \alpha}{2.0 + \left(\mathsf{fma}\left(2, i, \alpha\right) + \beta\right)}, 1.0\right)}{2.0}}\]
    3. Using strategy rm

    4. Applied add-exp-log7.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}, \frac{\beta - \alpha}{2.0 + \left(\mathsf{fma}\left(2, i, \alpha\right) + \beta\right)}, 1.0\right)\right)}}}{2.0}\]

    5. Simplified7.1

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right)\right)}}}{2.0}\]
    6. Using strategy rm

    7. Applied clear-num7.1

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right)\right)}}{2.0}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube7.1

      \[\leadsto \frac{e^{\log \color{blue}{\left(\sqrt[3]{\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right) \cdot \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right)\right) \cdot \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right)}\right)}}}{2.0}\]

    if 8.434843859125862e+184 < alpha

    1. Initial program 63.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Simplified49.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}, \frac{\beta - \alpha}{2.0 + \left(\mathsf{fma}\left(2, i, \alpha\right) + \beta\right)}, 1.0\right)}{2.0}}\]
    3. Using strategy rm

    4. Applied add-exp-log49.9

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}, \frac{\beta - \alpha}{2.0 + \left(\mathsf{fma}\left(2, i, \alpha\right) + \beta\right)}, 1.0\right)\right)}}}{2.0}\]

    5. Simplified49.9

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}, 1.0\right)\right)}}}{2.0}\]

    6. Taylor expanded around inf 43.2

      \[\leadsto \frac{e^{\log \color{blue}{\left(\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}\right)}}}{2.0}\]

    7. Simplified43.2

      \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)\right)}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.434843859125862 \cdot 10^{+184}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right) \cdot \mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}, \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1.0\right)\right)}\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))