Average Error: 3.7 → 2.6
Time: 21.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.4427574797641429 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\alpha \le 6.9212401795544938 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\alpha \le 9.03381831342100199 \cdot 10^{202}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}\;\alpha \le 2.4427574797641429 \cdot 10^{77}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{elif}\;\alpha \le 6.9212401795544938 \cdot 10^{125}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{elif}\;\alpha \le 9.03381831342100199 \cdot 10^{202}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r188550 = alpha;
        double r188551 = beta;
        double r188552 = r188550 + r188551;
        double r188553 = r188551 * r188550;
        double r188554 = r188552 + r188553;
        double r188555 = 1.0;
        double r188556 = r188554 + r188555;
        double r188557 = 2.0;
        double r188558 = r188557 * r188555;
        double r188559 = r188552 + r188558;
        double r188560 = r188556 / r188559;
        double r188561 = r188560 / r188559;
        double r188562 = r188559 + r188555;
        double r188563 = r188561 / r188562;
        return r188563;
}
↓
double f(double alpha, double beta) {
        double r188564 = alpha;
        double r188565 = 2.442757479764143e+77;
        bool r188566 = r188564 <= r188565;
        double r188567 = beta;
        double r188568 = 1.0;
        double r188569 = fma(r188564, r188567, r188568);
        double r188570 = r188564 + r188569;
        double r188571 = r188567 + r188570;
        double r188572 = r188564 + r188567;
        double r188573 = 2.0;
        double r188574 = r188573 * r188568;
        double r188575 = r188572 + r188574;
        double r188576 = sqrt(r188575);
        double r188577 = r188571 / r188576;
        double r188578 = fma(r188568, r188573, r188572);
        double r188579 = r188577 / r188578;
        double r188580 = r188579 / r188576;
        double r188581 = r188575 + r188568;
        double r188582 = r188580 / r188581;
        double r188583 = 6.921240179554494e+125;
        bool r188584 = r188564 <= r188583;
        double r188585 = 1.0;
        double r188586 = 2.0;
        double r188587 = pow(r188564, r188586);
        double r188588 = r188585 / r188587;
        double r188589 = r188585 / r188564;
        double r188590 = r188568 * r188589;
        double r188591 = r188585 - r188590;
        double r188592 = fma(r188573, r188588, r188591);
        double r188593 = r188592 / r188575;
        double r188594 = r188593 / r188581;
        double r188595 = 9.033818313421002e+202;
        bool r188596 = r188564 <= r188595;
        double r188597 = r188567 * r188564;
        double r188598 = r188572 + r188597;
        double r188599 = r188598 + r188568;
        double r188600 = sqrt(r188599);
        double r188601 = r188600 / r188576;
        double r188602 = r188601 / r188576;
        double r188603 = fma(r188573, r188568, r188568);
        double r188604 = r188572 + r188603;
        double r188605 = r188604 / r188600;
        double r188606 = r188605 * r188578;
        double r188607 = r188602 / r188606;
        double r188608 = r188596 ? r188607 : r188594;
        double r188609 = r188584 ? r188594 : r188608;
        double r188610 = r188566 ? r188582 : r188609;
        return r188610;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if alpha < 2.442757479764143e+77

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt1.0

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

    4. Applied associate-/r*0.5

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

    5. Simplified0.5

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

    if 2.442757479764143e+77 < alpha < 6.921240179554494e+125 or 9.033818313421002e+202 < alpha

    1. Initial program 14.7

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

    2. Taylor expanded around inf 7.9

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

    3. Simplified7.9

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

    if 6.921240179554494e+125 < alpha < 9.033818313421002e+202

    1. Initial program 10.5

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

    3. Applied add-sqr-sqrt10.6

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

    4. Applied add-sqr-sqrt10.7

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

    5. Applied add-sqr-sqrt10.6

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

    6. Applied times-frac10.6

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

    7. Applied times-frac10.7

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

    8. Applied associate-/l*10.7

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

    9. Simplified10.5

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.4427574797641429 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\alpha \le 6.9212401795544938 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\alpha \le 9.03381831342100199 \cdot 10^{202}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))