Average Error: 3.5 → 1.4
Time: 1.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)} \le 0.087775401979634876115099473281588871032:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]
\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}\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)} \le 0.087775401979634876115099473281588871032:\\
\;\;\;\;\sqrt{\frac{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}}\\

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

\end{array}
double f(double alpha, double beta) {
        double r5107801 = alpha;
        double r5107802 = beta;
        double r5107803 = r5107801 + r5107802;
        double r5107804 = r5107802 * r5107801;
        double r5107805 = r5107803 + r5107804;
        double r5107806 = 1.0;
        double r5107807 = r5107805 + r5107806;
        double r5107808 = 2.0;
        double r5107809 = r5107808 * r5107806;
        double r5107810 = r5107803 + r5107809;
        double r5107811 = r5107807 / r5107810;
        double r5107812 = r5107811 / r5107810;
        double r5107813 = r5107810 + r5107806;
        double r5107814 = r5107812 / r5107813;
        return r5107814;
}
↓
double f(double alpha, double beta) {
        double r5107815 = 1.0;
        double r5107816 = alpha;
        double r5107817 = beta;
        double r5107818 = r5107816 * r5107817;
        double r5107819 = r5107817 + r5107816;
        double r5107820 = r5107818 + r5107819;
        double r5107821 = r5107815 + r5107820;
        double r5107822 = 2.0;
        double r5107823 = r5107822 * r5107815;
        double r5107824 = r5107823 + r5107819;
        double r5107825 = r5107821 / r5107824;
        double r5107826 = r5107825 / r5107824;
        double r5107827 = r5107815 + r5107824;
        double r5107828 = r5107826 / r5107827;
        double r5107829 = 0.08777540197963488;
        bool r5107830 = r5107828 <= r5107829;
        double r5107831 = sqrt(r5107824);
        double r5107832 = r5107821 / r5107831;
        double r5107833 = r5107832 / r5107831;
        double r5107834 = r5107833 / r5107824;
        double r5107835 = r5107834 / r5107827;
        double r5107836 = sqrt(r5107835);
        double r5107837 = sqrt(r5107828);
        double r5107838 = r5107836 * r5107837;
        double r5107839 = r5107822 / r5107816;
        double r5107840 = r5107839 / r5107816;
        double r5107841 = 1.0;
        double r5107842 = r5107815 / r5107816;
        double r5107843 = r5107841 - r5107842;
        double r5107844 = r5107840 + r5107843;
        double r5107845 = r5107844 / r5107824;
        double r5107846 = r5107845 / r5107827;
        double r5107847 = r5107830 ? r5107838 : r5107846;
        return r5107847;
}

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 beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)) < 0.08777540197963488

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.2

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied associate-/r*0.2

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

    if 0.08777540197963488 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))

    1. Initial program 62.8

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

    2. Taylor expanded around inf 22.6

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

    3. Simplified22.6

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)} \le 0.087775401979634876115099473281588871032:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt{\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))