Octave 3.8, jcobi/1

Average Error: 16.8 → 3.8

Time: 4.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 0.00282165627097306351:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\alpha, 0.5, 1 - 0.5 \cdot \beta\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \end{array}\]

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 VAR;
	if ((beta <= 0.0028216562709730635)) {
		VAR = ((1.0 / fma(alpha, 0.5, (1.0 - (0.5 * beta)))) / 2.0);
	} else {
		VAR = (fma((beta - alpha), (1.0 / ((alpha + beta) + 2.0)), 1.0) / 2.0);
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if beta < 0.0028216562709730635

   1. Initial program 20.3

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

   3. Applied div-sub 20.3

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

   4. Applied associate-+l- 20.3

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

   6. Applied flip3-- 20.3

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

   7. Applied frac-sub 20.3

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

   8. Simplified 20.3

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

   9. Simplified 20.3

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

   11. Applied clear-num 20.3

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

   12. Taylor expanded around 0 0.7

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(0.5 \cdot \alpha + 1\right) - 0.5 \cdot \beta}}}{2}\]

   13. Simplified 0.7

       \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\alpha, 0.5, 1 - 0.5 \cdot \beta\right)}}}{2}\]

   if 0.0028216562709730635 < beta

   1. Initial program 9.8

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

   3. Applied div-inv 9.8

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

   4. Applied fma-def 9.8

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 0.00282165627097306351:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\alpha, 0.5, 1 - 0.5 \cdot \beta\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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))