Octave 3.8, jcobi/1

Average Error: 16.2 → 6.1

Time: 5.8s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 15329950251265012:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

double code(double alpha, double beta) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}

↓

double code(double alpha, double beta) {
	double temp;
	if ((alpha <= 15329950251265012.0)) {
		temp = (exp((log((pow((beta / ((alpha + beta) + 2.0)), 3.0) - pow(((alpha / ((alpha + beta) + 2.0)) - 1.0), 3.0))) - log((((beta / ((alpha + beta) + 2.0)) * (beta / ((alpha + beta) + 2.0))) + ((((alpha / ((alpha + beta) + 2.0)) - 1.0) * ((alpha / ((alpha + beta) + 2.0)) - 1.0)) + ((beta / ((alpha + beta) + 2.0)) * ((alpha / ((alpha + beta) + 2.0)) - 1.0))))))) / 2.0);
	} else {
		temp = (((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);
	}
	return temp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 15329950251265012.0

   1. Initial program 0.5

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

   3. Applied div-sub 0.5

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

   4. Applied associate-+l- 0.5

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

   6. Applied add-exp-log 0.5

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

   8. Applied flip3-- 0.5

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

   9. Applied log-div 0.5

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

   if 15329950251265012.0 < alpha

   1. Initial program 49.9

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

   3. Applied div-sub 49.9

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

   4. Applied associate-+l- 48.4

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

   5. Taylor expanded around inf 18.2

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

   6. Simplified 18.2

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

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 15329950251265012:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))