Average Error: 3.6 → 1.1
Time: 4.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.92589074825260795 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]
\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}\;\alpha \le 2.92589074825260795 \cdot 10^{159}:\\
\;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\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 <= 2.925890748252608e+159)) {
		VAR = ((((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) * (1.0 / ((alpha + beta) + (2.0 * 1.0)))) / (alpha + (beta + 3.0)));
	} else {
		VAR = (1.0 / (fma(1.0, 2.0, (alpha + beta)) * (2.0 + ((beta / alpha) + (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 alpha < 2.925890748252608e+159

    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. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    5. Applied div-inv1.3

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

    if 2.925890748252608e+159 < 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. Taylor expanded around 0 15.8

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

    3. Simplified15.8

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

    5. Applied clear-num16.0

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

    6. Simplified16.0

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

    7. Taylor expanded around inf 0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.92589074825260795 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))