Average Error: 3.6 → 2.4
Time: 2.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.0124820129180543 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{1}}}{2 \cdot \left(\alpha \cdot \sqrt{0.333333333333333315}\right) + \left(2 \cdot \frac{\beta}{\sqrt{3}} + 4 \cdot \sqrt{3}\right)}\\ \end{array}\]
\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}\;\alpha \le 6.0124820129180543 \cdot 10^{212}:\\
\;\;\;\;\frac{\frac{\sqrt{\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}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt{\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}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{1}}}{2 \cdot \left(\alpha \cdot \sqrt{0.333333333333333315}\right) + \left(2 \cdot \frac{\beta}{\sqrt{3}} + 4 \cdot \sqrt{3}\right)}\\

\end{array}
double code(double alpha, double beta) {
	return ((((((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));
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 6.012482012918054e+212)) {
		VAR = ((sqrt(((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0)))) / sqrt(((alpha + beta) + (2.0 * 1.0)))) / ((((alpha + beta) + (2.0 * 1.0)) + 1.0) / (sqrt(((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0)))) / sqrt(((alpha + beta) + (2.0 * 1.0))))));
	} else {
		VAR = ((1.0 / (sqrt((((alpha + beta) + (2.0 * 1.0)) + 1.0)) / 1.0)) / ((2.0 * (alpha * sqrt(0.3333333333333333))) + ((2.0 * (beta / sqrt(3.0))) + (4.0 * sqrt(3.0)))));
	}
	return VAR;
}

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 < 6.012482012918054e+212

    1. Initial program 2.0

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

    3. Applied add-sqr-sqrt2.5

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

    4. Applied add-sqr-sqrt2.1

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

    5. Applied times-frac2.1

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

    6. Applied associate-/l*2.1

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

    if 6.012482012918054e+212 < alpha

    1. Initial program 17.3

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

    3. Applied *-un-lft-identity17.3

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

    4. Applied *-un-lft-identity17.3

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

    5. Applied times-frac17.3

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

    6. Applied associate-/l*17.3

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

    8. Applied *-un-lft-identity17.3

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

    9. Applied *-un-lft-identity17.3

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

    10. Applied *-un-lft-identity17.3

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

    11. Applied times-frac17.3

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

    12. Applied times-frac17.3

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

    13. Applied add-sqr-sqrt17.3

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

    14. Applied times-frac17.3

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

    15. Applied associate-/r*17.3

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

    16. Simplified17.3

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

    17. Taylor expanded around 0 4.4

      \[\leadsto \frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{1}}}{\color{blue}{2 \cdot \left(\alpha \cdot \sqrt{0.333333333333333315}\right) + \left(2 \cdot \frac{\beta}{\sqrt{3}} + 4 \cdot \sqrt{3}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.0124820129180543 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{1}}}{2 \cdot \left(\alpha \cdot \sqrt{0.333333333333333315}\right) + \left(2 \cdot \frac{\beta}{\sqrt{3}} + 4 \cdot \sqrt{3}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))