Average Error: 24.0 → 11.4
Time: 21.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r135599 = alpha;
        double r135600 = beta;
        double r135601 = r135599 + r135600;
        double r135602 = r135600 - r135599;
        double r135603 = r135601 * r135602;
        double r135604 = 2.0;
        double r135605 = i;
        double r135606 = r135604 * r135605;
        double r135607 = r135601 + r135606;
        double r135608 = r135603 / r135607;
        double r135609 = r135607 + r135604;
        double r135610 = r135608 / r135609;
        double r135611 = 1.0;
        double r135612 = r135610 + r135611;
        double r135613 = r135612 / r135604;
        return r135613;
}
↓
double f(double alpha, double beta, double i) {
        double r135614 = alpha;
        double r135615 = 4.195090868122503e+103;
        bool r135616 = r135614 <= r135615;
        double r135617 = 1.0;
        double r135618 = i;
        double r135619 = 2.0;
        double r135620 = beta;
        double r135621 = r135614 + r135620;
        double r135622 = fma(r135618, r135619, r135621);
        double r135623 = r135622 / r135621;
        double r135624 = r135620 - r135614;
        double r135625 = r135623 / r135624;
        double r135626 = r135617 / r135625;
        double r135627 = r135619 * r135618;
        double r135628 = r135621 + r135627;
        double r135629 = r135628 + r135619;
        double r135630 = r135617 / r135629;
        double r135631 = 1.0;
        double r135632 = fma(r135626, r135630, r135631);
        double r135633 = 3.0;
        double r135634 = pow(r135632, r135633);
        double r135635 = cbrt(r135634);
        double r135636 = pow(r135635, r135633);
        double r135637 = cbrt(r135636);
        double r135638 = r135637 / r135619;
        double r135639 = r135617 / r135614;
        double r135640 = 8.0;
        double r135641 = pow(r135614, r135633);
        double r135642 = r135617 / r135641;
        double r135643 = r135640 * r135642;
        double r135644 = 4.0;
        double r135645 = 2.0;
        double r135646 = pow(r135614, r135645);
        double r135647 = r135617 / r135646;
        double r135648 = r135644 * r135647;
        double r135649 = r135643 - r135648;
        double r135650 = fma(r135619, r135639, r135649);
        double r135651 = r135650 / r135619;
        double r135652 = r135616 ? r135638 : r135651;
        return r135652;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.195090868122503e+103

    1. Initial program 13.8

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

    3. Applied clear-num13.8

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

    4. Simplified3.2

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

    6. Applied div-inv3.2

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

    7. Applied fma-def3.2

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

    9. Applied add-cbrt-cube3.2

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

    10. Simplified3.2

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

    12. Applied add-cbrt-cube3.2

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

    13. Simplified3.2

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

    if 4.195090868122503e+103 < alpha

    1. Initial program 59.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Taylor expanded around inf 39.6

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

    3. Simplified39.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))