Average Error: 23.3 → 11.5
Time: 22.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 1.755371486194466 \cdot 10^{248}:\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}} + 1}{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 r138720 = alpha;
        double r138721 = beta;
        double r138722 = r138720 + r138721;
        double r138723 = r138721 - r138720;
        double r138724 = r138722 * r138723;
        double r138725 = 2.0;
        double r138726 = i;
        double r138727 = r138725 * r138726;
        double r138728 = r138722 + r138727;
        double r138729 = r138724 / r138728;
        double r138730 = r138728 + r138725;
        double r138731 = r138729 / r138730;
        double r138732 = 1.0;
        double r138733 = r138731 + r138732;
        double r138734 = r138733 / r138725;
        return r138734;
}
↓
double f(double alpha, double beta, double i) {
        double r138735 = alpha;
        double r138736 = 1.7553714861944655e+248;
        bool r138737 = r138735 <= r138736;
        double r138738 = beta;
        double r138739 = r138735 + r138738;
        double r138740 = 1.0;
        double r138741 = 2.0;
        double r138742 = i;
        double r138743 = r138741 * r138742;
        double r138744 = r138739 + r138743;
        double r138745 = r138744 + r138741;
        double r138746 = sqrt(r138745);
        double r138747 = r138740 / r138746;
        double r138748 = fma(r138742, r138741, r138739);
        double r138749 = r138738 - r138735;
        double r138750 = cbrt(r138745);
        double r138751 = r138750 * r138750;
        double r138752 = sqrt(r138751);
        double r138753 = r138749 / r138752;
        double r138754 = sqrt(r138750);
        double r138755 = r138753 / r138754;
        double r138756 = r138748 / r138755;
        double r138757 = r138747 / r138756;
        double r138758 = r138739 * r138757;
        double r138759 = 1.0;
        double r138760 = r138758 + r138759;
        double r138761 = r138760 / r138741;
        double r138762 = r138740 / r138735;
        double r138763 = 8.0;
        double r138764 = 3.0;
        double r138765 = pow(r138735, r138764);
        double r138766 = r138740 / r138765;
        double r138767 = r138763 * r138766;
        double r138768 = 4.0;
        double r138769 = 2.0;
        double r138770 = pow(r138735, r138769);
        double r138771 = r138740 / r138770;
        double r138772 = r138768 * r138771;
        double r138773 = r138767 - r138772;
        double r138774 = fma(r138741, r138762, r138773);
        double r138775 = r138774 / r138741;
        double r138776 = r138737 ? r138761 : r138775;
        return r138776;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.7553714861944655e+248

    1. Initial program 20.6

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

    3. Applied *-un-lft-identity20.6

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

    4. Applied *-un-lft-identity20.6

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

    5. Applied times-frac9.5

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

    6. Applied times-frac9.4

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

    7. Simplified9.4

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

    8. Simplified9.5

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

    10. Applied add-sqr-sqrt9.5

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

    11. Applied *-un-lft-identity9.5

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

    12. Applied times-frac9.5

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

    13. Applied associate-/l*9.5

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

    15. Applied add-cube-cbrt9.5

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

    16. Applied sqrt-prod9.5

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

    17. Applied associate-/r*9.5

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

    if 1.7553714861944655e+248 < 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. Taylor expanded around inf 40.5

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

    3. Simplified40.5

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\frac{\beta - \alpha}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))