Average Error: 16.0 → 0.2
Time: 4.6s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999999999912373:\\ \;\;\;\;\frac{\beta + 1}{\alpha} - \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, 3, \frac{2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\
\mathbf{if}\;t_0 \leq -0.9999999999912373:\\
\;\;\;\;\frac{\beta + 1}{\alpha} - \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, 3, \frac{2}{\alpha \cdot \alpha}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0 + 1}{2}\\


\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) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999999999912373)
     (-
      (/ (+ beta 1.0) alpha)
      (fma (/ beta (* alpha alpha)) 3.0 (/ 2.0 (* alpha alpha))))
     (/ (+ t_0 1.0) 2.0))))
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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999999999912373) {
		tmp = ((beta + 1.0) / alpha) - fma((beta / (alpha * alpha)), 3.0, (2.0 / (alpha * alpha)));
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	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)) < -0.999999999991237343

    1. Initial program 60.2

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

    2. Simplified60.2

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

    3. Taylor expanded in alpha around inf 2.5

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

    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 beta around 0 0.1

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

    6. Simplified0.1

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

    if -0.999999999991237343 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999912373:\\ \;\;\;\;\frac{\beta + 1}{\alpha} - \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, 3, \frac{2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))