Average Error: 3.8 → 4.6
Time: 37.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.46570123069215477 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\frac{-\left(\left(\mathsf{fma}\left(\alpha, \beta, \beta\right) + \alpha\right) + 1\right)}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5 + 0.25 \cdot \beta\right)}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{1} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.46570123069215477 \cdot 10^{113}:\\
\;\;\;\;\frac{\frac{\frac{-\left(\left(\mathsf{fma}\left(\alpha, \beta, \beta\right) + \alpha\right) + 1\right)}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5 + 0.25 \cdot \beta\right)}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{1} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\

\end{array}
double code(double alpha, double beta) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0))));
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 1.4657012306921548e+113)) {
		VAR = ((double) (((double) (((double) (((double) -(((double) (((double) (((double) fma(alpha, beta, beta)) + alpha)) + 1.0)))) / ((double) -(((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0))));
	} else {
		VAR = ((double) (((double) fma(0.25, alpha, ((double) (0.5 + ((double) (0.25 * beta)))))) / ((double) (((double) (((double) (((double) (alpha + beta)) + ((double) fma(2.0, 1.0, 1.0)))) / 1.0)) * ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))))));
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.4657012306921548e+113

    1. Initial program 0.6

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

    3. Applied frac-2neg0.6

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

    4. Simplified0.6

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

    if 1.4657012306921548e+113 < alpha

    1. Initial program 15.1

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

    3. Applied frac-2neg15.1

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

    4. Simplified15.1

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

    6. Applied div-inv15.1

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

    7. Applied associate-/l*17.7

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

    8. Simplified17.7

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

    9. Taylor expanded around 0 18.8

      \[\leadsto \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{1} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\]

    10. Simplified18.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25, \alpha, 0.5 + 0.25 \cdot \beta\right)}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{1} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.46570123069215477 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\frac{-\left(\left(\mathsf{fma}\left(\alpha, \beta, \beta\right) + \alpha\right) + 1\right)}{-\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5 + 0.25 \cdot \beta\right)}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{1} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))