Average Error: 3.8 → 1.3
Time: 9.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.38281142000122207 \cdot 10^{164}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]
\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 8.38281142000122207 \cdot 10^{164}:\\
\;\;\;\;\sqrt{\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}} \cdot \frac{\sqrt{\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}}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\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 ((beta <= 8.382811420001222e+164)) {
		VAR = (sqrt((((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0)))) * (sqrt((((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0)))) / (beta + (alpha + fma(1.0, 2.0, 1.0)))));
	} else {
		VAR = (1.0 / ((2.0 + ((beta / alpha) + (alpha / beta))) * fma(1.0, 2.0, (alpha + beta))));
	}
	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 beta < 8.382811420001222e+164

    1. Initial program 1.3

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

    3. Applied *-un-lft-identity1.3

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

    4. Applied add-sqr-sqrt1.4

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

    5. Applied times-frac1.4

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

    6. Simplified1.4

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

    7. Simplified1.4

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

    if 8.382811420001222e+164 < beta

    1. Initial program 16.9

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

    3. Applied clear-num16.9

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

    4. Simplified16.9

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

    6. Applied clear-num17.0

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

    7. Simplified17.0

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

    8. Taylor expanded around inf 0.7

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.38281142000122207 \cdot 10^{164}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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)))