Average Error: 3.4 → 2.6
Time: 14.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.9207371840512 \cdot 10^{197}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\frac{\mathsf{fma}\left(\beta, \sqrt{0.5}, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 2.9207371840512 \cdot 10^{197}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\frac{\mathsf{fma}\left(\beta, \sqrt{0.5}, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r117626 = alpha;
        double r117627 = beta;
        double r117628 = r117626 + r117627;
        double r117629 = r117627 * r117626;
        double r117630 = r117628 + r117629;
        double r117631 = 1.0;
        double r117632 = r117630 + r117631;
        double r117633 = 2.0;
        double r117634 = r117633 * r117631;
        double r117635 = r117628 + r117634;
        double r117636 = r117632 / r117635;
        double r117637 = r117636 / r117635;
        double r117638 = r117635 + r117631;
        double r117639 = r117637 / r117638;
        return r117639;
}
↓
double f(double alpha, double beta) {
        double r117640 = beta;
        double r117641 = 2.9207371840511997e+197;
        bool r117642 = r117640 <= r117641;
        double r117643 = alpha;
        double r117644 = r117643 + r117640;
        double r117645 = r117640 * r117643;
        double r117646 = r117644 + r117645;
        double r117647 = 1.0;
        double r117648 = r117646 + r117647;
        double r117649 = 2.0;
        double r117650 = fma(r117647, r117649, r117644);
        double r117651 = r117648 / r117650;
        double r117652 = r117649 * r117647;
        double r117653 = r117644 - r117652;
        double r117654 = r117651 / r117653;
        double r117655 = r117654 / r117650;
        double r117656 = r117655 * r117653;
        double r117657 = r117644 + r117652;
        double r117658 = r117657 + r117647;
        double r117659 = r117656 / r117658;
        double r117660 = 1.0;
        double r117661 = sqrt(r117657);
        double r117662 = r117660 / r117661;
        double r117663 = sqrt(r117658);
        double r117664 = r117662 / r117663;
        double r117665 = 0.5;
        double r117666 = sqrt(r117665);
        double r117667 = 0.75;
        double r117668 = r117643 * r117666;
        double r117669 = r117647 * r117666;
        double r117670 = fma(r117667, r117668, r117669);
        double r117671 = 0.125;
        double r117672 = r117640 / r117666;
        double r117673 = r117671 * r117672;
        double r117674 = r117670 - r117673;
        double r117675 = fma(r117640, r117666, r117674);
        double r117676 = r117675 / r117663;
        double r117677 = r117676 / r117650;
        double r117678 = r117664 * r117677;
        double r117679 = r117642 ? r117659 : r117678;
        return r117679;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.9207371840511997e+197

    1. Initial program 1.6

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

    3. Applied flip-+2.4

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

    4. Applied associate-/r/2.4

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

    5. Simplified1.6

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

    if 2.9207371840511997e+197 < beta

    1. Initial program 16.3

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

    3. Applied add-sqr-sqrt16.3

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

    4. Applied *-un-lft-identity16.3

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

    5. Applied add-sqr-sqrt16.3

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

    6. Applied *-un-lft-identity16.3

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

    7. Applied times-frac16.3

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

    8. Applied times-frac16.3

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

    9. Applied times-frac16.3

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

    10. Simplified16.3

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

    11. Simplified16.3

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

    12. Taylor expanded around 0 10.4

      \[\leadsto \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\]

    13. Simplified10.4

      \[\leadsto \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(\beta, \sqrt{0.5}, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.9207371840512 \cdot 10^{197}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\frac{\mathsf{fma}\left(\beta, \sqrt{0.5}, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))