Average Error: 3.8 → 1.2
Time: 8.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1125280590322559 \cdot 10^{161}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}\;\beta \le 1.1125280590322559 \cdot 10^{161}:\\
\;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\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 temp;
	if ((beta <= 1.1125280590322559e+161)) {
		temp = (log1p(expm1((1.0 / ((fma(1.0, 2.0, (alpha + beta)) / (beta + (alpha + fma(alpha, beta, 1.0)))) * fma(1.0, 2.0, (alpha + beta)))))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0));
	} else {
		temp = (log1p(expm1((1.0 / (2.0 + ((beta / alpha) + (alpha / beta)))))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0));
	}
	return temp;
}

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 < 1.1125280590322559e+161

    1. Initial program 1.4

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

    3. Applied log1p-expm1-u1.4

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity1.4

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Applied *-un-lft-identity1.4

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Applied times-frac1.4

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    8. Applied associate-/l*1.4

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\frac{1}{1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    9. Simplified1.4

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{1}}{\color{blue}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    if 1.1125280590322559e+161 < beta

    1. Initial program 16.2

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

    3. Applied log1p-expm1-u16.2

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity16.2

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Applied *-un-lft-identity16.2

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Applied times-frac16.2

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    8. Applied associate-/l*16.2

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\frac{1}{1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    9. Simplified16.2

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{1}}{\color{blue}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    10. Taylor expanded around inf 0.1

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{1}}{\color{blue}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1125280590322559 \cdot 10^{161}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}{\beta + \left(\alpha + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))