Average Error: 16.1 → 7.2
Time: 4.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.69809026129430447 \cdot 10^{38}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \frac{\left(\alpha + \beta\right) + 2}{\beta} \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \frac{\left(\alpha + \beta\right) + 2}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \le 1.1380605375375739 \cdot 10^{53}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.2936649981807928 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}, \left(\alpha + \beta\right) - 2, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.69809026129430447 \cdot 10^{38}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \frac{\left(\alpha + \beta\right) + 2}{\beta} \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \frac{\left(\alpha + \beta\right) + 2}{\beta}}}{2}\\

\mathbf{elif}\;\alpha \le 1.1380605375375739 \cdot 10^{53}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\mathbf{elif}\;\alpha \le 5.2936649981807928 \cdot 10^{65}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}, \left(\alpha + \beta\right) - 2, -1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double code(double alpha, double beta) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 1.6980902612943045e+38)) {
		VAR = (((fma((alpha / ((alpha + beta) + 2.0)), (alpha / ((alpha + beta) + 2.0)), fma(1.0, 1.0, ((alpha / ((alpha + beta) + 2.0)) * 1.0))) - ((((alpha + beta) + 2.0) / beta) * (pow((alpha / ((alpha + beta) + 2.0)), 3.0) - pow(1.0, 3.0)))) / (fma((alpha / ((alpha + beta) + 2.0)), (alpha / ((alpha + beta) + 2.0)), fma(1.0, 1.0, ((alpha / ((alpha + beta) + 2.0)) * 1.0))) * (((alpha + beta) + 2.0) / beta))) / 2.0);
	} else {
		double VAR_1;
		if ((alpha <= 1.138060537537574e+53)) {
			VAR_1 = (((beta / ((alpha + beta) + 2.0)) - fma(4.0, (1.0 / pow(alpha, 2.0)), -fma(2.0, (1.0 / alpha), (8.0 * (1.0 / pow(alpha, 3.0)))))) / 2.0);
		} else {
			double VAR_2;
			if ((alpha <= 5.293664998180793e+65)) {
				VAR_2 = (((beta / ((alpha + beta) + 2.0)) - fma((alpha / (((alpha + beta) * (alpha + beta)) - (2.0 * 2.0))), ((alpha + beta) - 2.0), -1.0)) / 2.0);
			} else {
				VAR_2 = (((beta / ((alpha + beta) + 2.0)) - fma(4.0, (1.0 / pow(alpha, 2.0)), -fma(2.0, (1.0 / alpha), (8.0 * (1.0 / pow(alpha, 3.0)))))) / 2.0);
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if alpha < 1.6980902612943045e+38

    1. Initial program 1.8

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

    3. Applied div-sub1.8

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

    4. Applied associate-+l-1.8

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

    6. Applied flip3--1.8

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

    7. Applied clear-num1.8

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

    8. Applied frac-sub1.9

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

    9. Simplified1.9

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

    10. Simplified1.9

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

    if 1.6980902612943045e+38 < alpha < 1.138060537537574e+53 or 5.293664998180793e+65 < alpha

    1. Initial program 51.3

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

    3. Applied div-sub51.3

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around inf 18.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    6. Simplified18.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    if 1.138060537537574e+53 < alpha < 5.293664998180793e+65

    1. Initial program 44.6

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

    3. Applied div-sub44.6

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

    4. Applied associate-+l-44.2

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

    6. Applied flip-+44.2

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

    7. Applied associate-/r/43.9

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

    8. Applied fma-neg43.9

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.69809026129430447 \cdot 10^{38}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \frac{\left(\alpha + \beta\right) + 2}{\beta} \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}, \frac{\alpha}{\left(\alpha + \beta\right) + 2}, \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) \cdot \frac{\left(\alpha + \beta\right) + 2}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \le 1.1380605375375739 \cdot 10^{53}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.2936649981807928 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}, \left(\alpha + \beta\right) - 2, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))