Average Error: 23.7 → 10.9
Time: 12.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}} \cdot {\left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2\right)}^{\frac{-3}{4}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 1.5371929371909745 \cdot 10^{209}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}} \cdot {\left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2\right)}^{\frac{-3}{4}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r90636 = alpha;
        double r90637 = beta;
        double r90638 = r90636 + r90637;
        double r90639 = r90637 - r90636;
        double r90640 = r90638 * r90639;
        double r90641 = 2.0;
        double r90642 = i;
        double r90643 = r90641 * r90642;
        double r90644 = r90638 + r90643;
        double r90645 = r90640 / r90644;
        double r90646 = r90644 + r90641;
        double r90647 = r90645 / r90646;
        double r90648 = 1.0;
        double r90649 = r90647 + r90648;
        double r90650 = r90649 / r90641;
        return r90650;
}
↓
double f(double alpha, double beta, double i) {
        double r90651 = alpha;
        double r90652 = 1.5371929371909745e+209;
        bool r90653 = r90651 <= r90652;
        double r90654 = beta;
        double r90655 = r90651 + r90654;
        double r90656 = 2.0;
        double r90657 = i;
        double r90658 = fma(r90656, r90657, r90655);
        double r90659 = r90658 + r90656;
        double r90660 = sqrt(r90659);
        double r90661 = sqrt(r90660);
        double r90662 = r90655 / r90661;
        double r90663 = -0.75;
        double r90664 = pow(r90659, r90663);
        double r90665 = r90662 * r90664;
        double r90666 = r90654 - r90651;
        double r90667 = r90666 / r90658;
        double r90668 = 1.0;
        double r90669 = fma(r90665, r90667, r90668);
        double r90670 = r90669 / r90656;
        double r90671 = 8.0;
        double r90672 = 3.0;
        double r90673 = pow(r90651, r90672);
        double r90674 = r90671 / r90673;
        double r90675 = r90656 / r90651;
        double r90676 = 4.0;
        double r90677 = r90651 * r90651;
        double r90678 = r90676 / r90677;
        double r90679 = r90675 - r90678;
        double r90680 = r90674 + r90679;
        double r90681 = r90680 / r90656;
        double r90682 = r90653 ? r90670 : r90681;
        return r90682;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.5371929371909745e+209

    1. Initial program 18.9

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

    2. Simplified7.4

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

    4. Applied add-sqr-sqrt7.4

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

    5. Applied associate-/r*7.4

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

    7. Applied add-sqr-sqrt7.4

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

    8. Applied sqrt-prod7.5

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

    9. Applied div-inv7.5

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

    10. Applied times-frac7.4

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

    12. Applied pow1/27.4

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

    13. Applied sqrt-pow17.4

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

    14. Applied pow1/27.4

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

    15. Applied pow-flip7.4

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

    16. Applied pow-div7.4

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

    17. Simplified7.4

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

    if 1.5371929371909745e+209 < alpha

    1. Initial program 64.0

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

    2. Simplified50.3

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

    3. Taylor expanded around inf 40.5

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

    4. Simplified40.5

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}} \cdot {\left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2\right)}^{\frac{-3}{4}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))