Average Error: 3.8 → 0.2
Time: 8.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.603857948514167 \cdot 10^{+128}:\\ \;\;\;\;{\left(\frac{\left(\beta + \alpha\right) + 3}{\left(\beta + \alpha\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)} \cdot {\left(\alpha + \left(\beta + 2\right)\right)}^{2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) - \mathsf{fma}\left(2, \frac{\alpha}{\beta} \cdot \frac{\alpha}{\beta}, \mathsf{fma}\left(5, \frac{\alpha}{\beta \cdot \beta}, \frac{3}{\beta \cdot \beta}\right)\right)}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.603857948514167e+128)
   (pow
    (*
     (/ (+ (+ beta alpha) 3.0) (+ (+ beta alpha) (fma alpha beta 1.0)))
     (pow (+ alpha (+ beta 2.0)) 2.0))
    -1.0)
   (/
    (-
     (+ (/ 1.0 beta) (/ alpha beta))
     (fma
      2.0
      (* (/ alpha beta) (/ alpha beta))
      (fma 5.0 (/ alpha (* beta beta)) (/ 3.0 (* beta beta)))))
    (+ 1.0 (+ (+ beta alpha) 2.0)))))
double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.603857948514167e+128) {
		tmp = pow(((((beta + alpha) + 3.0) / ((beta + alpha) + fma(alpha, beta, 1.0))) * pow((alpha + (beta + 2.0)), 2.0)), -1.0);
	} else {
		tmp = (((1.0 / beta) + (alpha / beta)) - fma(2.0, ((alpha / beta) * (alpha / beta)), fma(5.0, (alpha / (beta * beta)), (3.0 / (beta * beta))))) / (1.0 + ((beta + alpha) + 2.0));
	}
	return tmp;
}

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end
↓
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.603857948514167e+128)
		tmp = Float64(Float64(Float64(Float64(beta + alpha) + 3.0) / Float64(Float64(beta + alpha) + fma(alpha, beta, 1.0))) * (Float64(alpha + Float64(beta + 2.0)) ^ 2.0)) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) - fma(2.0, Float64(Float64(alpha / beta) * Float64(alpha / beta)), fma(5.0, Float64(alpha / Float64(beta * beta)), Float64(3.0 / Float64(beta * beta))))) / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	end
	return tmp
end

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[alpha_, beta_] := If[LessEqual[beta, 4.603857948514167e+128], N[Power[N[(N[(N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(alpha / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(3.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\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}\;\beta \leq 4.603857948514167 \cdot 10^{+128}:\\
\;\;\;\;{\left(\frac{\left(\beta + \alpha\right) + 3}{\left(\beta + \alpha\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)} \cdot {\left(\alpha + \left(\beta + 2\right)\right)}^{2}\right)}^{-1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) - \mathsf{fma}\left(2, \frac{\alpha}{\beta} \cdot \frac{\alpha}{\beta}, \mathsf{fma}\left(5, \frac{\alpha}{\beta \cdot \beta}, \frac{3}{\beta \cdot \beta}\right)\right)}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\


\end{array}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.6038579485141671e128

    1. Initial program 0.1

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

    2. Applied egg-rr0.1

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

    if 4.6038579485141671e128 < beta

    1. Initial program 10.6

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

    2. Taylor expanded in beta around inf 5.9

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) - \left(2 \cdot \frac{{\alpha}^{2}}{{\beta}^{2}} + \left(5 \cdot \frac{\alpha}{{\beta}^{2}} + 3 \cdot \frac{1}{{\beta}^{2}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

    3. Simplified0.4

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) - \mathsf{fma}\left(2, \frac{\alpha}{\beta} \cdot \frac{\alpha}{\beta}, \mathsf{fma}\left(5, \frac{\alpha}{\beta \cdot \beta}, \frac{3}{\beta \cdot \beta}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.603857948514167 \cdot 10^{+128}:\\ \;\;\;\;{\left(\frac{\left(\beta + \alpha\right) + 3}{\left(\beta + \alpha\right) + \mathsf{fma}\left(\alpha, \beta, 1\right)} \cdot {\left(\alpha + \left(\beta + 2\right)\right)}^{2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) - \mathsf{fma}\left(2, \frac{\alpha}{\beta} \cdot \frac{\alpha}{\beta}, \mathsf{fma}\left(5, \frac{\alpha}{\beta \cdot \beta}, \frac{3}{\beta \cdot \beta}\right)\right)}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))