Average Error: 16.4 → 1.0
Time: 4.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999999999997027:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \frac{\mathsf{fma}\left(\beta, 3, \mathsf{fma}\left(\beta, \beta, 2\right)\right)}{\alpha \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t_0}{\alpha - \beta}}, -0.5, 0.5\right)\\ \end{array} \]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
t_0 := \left(\beta + \alpha\right) + 2\\
\mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999999999997027:\\
\;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \frac{\mathsf{fma}\left(\beta, 3, \mathsf{fma}\left(\beta, \beta, 2\right)\right)}{\alpha \cdot \alpha}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t_0}{\alpha - \beta}}, -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
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999999999997027)
     (-
      (+ (/ 1.0 alpha) (/ beta alpha))
      (/ (fma beta 3.0 (fma beta beta 2.0)) (* alpha alpha)))
     (fma (/ 1.0 (/ t_0 (- alpha beta))) -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 t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999999999997027) {
		tmp = ((1.0 / alpha) + (beta / alpha)) - (fma(beta, 3.0, fma(beta, beta, 2.0)) / (alpha * alpha));
	} else {
		tmp = fma((1.0 / (t_0 / (alpha - beta))), -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)) < -0.99999999999970268

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded in alpha around inf 2.8

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

    4. Simplified0.0

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

    5. Taylor expanded in alpha around 0 2.8

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

    6. Simplified2.8

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

    if -0.99999999999970268 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.3

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

    2. Simplified0.3

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

    3. Applied clear-num_binary640.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha - \beta}}}, -0.5, 0.5\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999997027:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \frac{\mathsf{fma}\left(\beta, 3, \mathsf{fma}\left(\beta, \beta, 2\right)\right)}{\alpha \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha - \beta}}, -0.5, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))