Average Error: 16.2 → 0.3
Time: 4.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999999989:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)\right), -0.5, 0.5\right)\\ \end{array} \]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999999989:\\
\;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)\right), -0.5, 0.5\right)\\


\end{array}

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999999999989)
   (-
    (+ (/ 1.0 alpha) (/ beta alpha))
    (+
     (/ 2.0 (* alpha alpha))
     (fma 3.0 (/ beta (* alpha alpha)) (* (/ beta alpha) (/ beta alpha)))))
   (fma (log1p (expm1 (/ (- alpha beta) (+ beta (+ alpha 2.0))))) -0.5 0.5)))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999999999989) {
		tmp = ((1.0 / alpha) + (beta / alpha)) - ((2.0 / (alpha * alpha)) + fma(3.0, (beta / (alpha * alpha)), ((beta / alpha) * (beta / alpha))));
	} else {
		tmp = fma(log1p(expm1((alpha - beta) / (beta + (alpha + 2.0)))), -0.5, 0.5);
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999999999999889

    1. Initial program 60.5

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

    2. Simplified60.5

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

    3. Taylor expanded in alpha around inf 2.9

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

    4. Simplified0.0

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

    if -0.99999999999999889 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.4

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

    2. Simplified0.4

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

    3. Applied log1p-expm1-u_binary640.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}\right)\right)}, -0.5, 0.5\right) \]

    4. Applied expm1-log1p-u_binary640.4

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}\right)\right)\right)}\right), -0.5, 0.5\right) \]

    5. Simplified0.4

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\alpha - \beta}{\beta + \left(2 + \alpha\right)}}\right)\right), -0.5, 0.5\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999999989:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)\right), -0.5, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))