Average Error: 3.6 → 1.4
Time: 10.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.12425572731382792 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 3.12425572731382792 \cdot 10^{160}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}\right) \cdot \sqrt{\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 r96707 = alpha;
        double r96708 = beta;
        double r96709 = r96707 + r96708;
        double r96710 = r96708 * r96707;
        double r96711 = r96709 + r96710;
        double r96712 = 1.0;
        double r96713 = r96711 + r96712;
        double r96714 = 2.0;
        double r96715 = r96714 * r96712;
        double r96716 = r96709 + r96715;
        double r96717 = r96713 / r96716;
        double r96718 = r96717 / r96716;
        double r96719 = r96716 + r96712;
        double r96720 = r96718 / r96719;
        return r96720;
}
↓
double f(double alpha, double beta) {
        double r96721 = alpha;
        double r96722 = 3.124255727313828e+160;
        bool r96723 = r96721 <= r96722;
        double r96724 = beta;
        double r96725 = r96721 + r96724;
        double r96726 = r96724 * r96721;
        double r96727 = r96725 + r96726;
        double r96728 = 1.0;
        double r96729 = r96727 + r96728;
        double r96730 = 2.0;
        double r96731 = r96730 * r96728;
        double r96732 = r96725 + r96731;
        double r96733 = r96729 / r96732;
        double r96734 = 3.0;
        double r96735 = pow(r96733, r96734);
        double r96736 = cbrt(r96735);
        double r96737 = r96736 / r96732;
        double r96738 = 3.0;
        double r96739 = r96724 + r96738;
        double r96740 = r96721 + r96739;
        double r96741 = r96737 / r96740;
        double r96742 = 1.0;
        double r96743 = sqrt(r96732);
        double r96744 = r96742 / r96743;
        double r96745 = r96742 / r96721;
        double r96746 = r96742 / r96724;
        double r96747 = r96745 + r96746;
        double r96748 = 2.0;
        double r96749 = pow(r96721, r96748);
        double r96750 = r96742 / r96749;
        double r96751 = r96747 - r96750;
        double r96752 = r96751 * r96743;
        double r96753 = r96744 / r96752;
        double r96754 = r96732 + r96728;
        double r96755 = r96753 / r96754;
        double r96756 = r96723 ? r96741 : r96755;
        return r96756;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.124255727313828e+160

    1. Initial program 1.2

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\alpha + \left(3 + \beta\right)}}\]

    3. Simplified1.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\alpha + \left(\beta + 3\right)}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube8.3

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

    6. Applied add-cbrt-cube18.0

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

    7. Applied cbrt-undiv18.0

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

    8. Simplified1.7

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

    if 3.124255727313828e+160 < alpha

    1. Initial program 15.8

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

    3. Applied add-sqr-sqrt15.9

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

    4. Applied *-un-lft-identity15.9

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

    5. Applied times-frac15.9

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

    6. Applied associate-/l*15.9

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

    7. Simplified15.9

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

    8. Taylor expanded around inf 0.1

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.12425572731382792 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}^{3}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))