Average Error: 3.6 → 0.1
Time: 11.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
↓
\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.744835924247743 \cdot 10^{+111}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t_0} \cdot \frac{1}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-1, \frac{\frac{\alpha}{-1 - \alpha} + \left(\frac{2}{-1 - \alpha} + \frac{\alpha + 1}{{\left(-1 - \alpha\right)}^{2}}\right)}{\beta}, \frac{-1}{-1 - \alpha}\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
(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
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.744835924247743e+111)
     (*
      (/ (+ alpha (+ beta (fma alpha beta 1.0))) t_0)
      (/ 1.0 (* t_0 (+ beta (+ alpha 3.0)))))
     (/
      (/
       1.0
       (*
        (fma
         -1.0
         (/
          (+
           (/ alpha (- -1.0 alpha))
           (+
            (/ 2.0 (- -1.0 alpha))
            (/ (+ alpha 1.0) (pow (- -1.0 alpha) 2.0))))
          beta)
         (/ -1.0 (- -1.0 alpha)))
        (+ beta (+ alpha 2.0))))
      (+ 1.0 (+ 2.0 (+ beta alpha)))))))
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 t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.744835924247743e+111) {
		tmp = ((alpha + (beta + fma(alpha, beta, 1.0))) / t_0) * (1.0 / (t_0 * (beta + (alpha + 3.0))));
	} else {
		tmp = (1.0 / (fma(-1.0, (((alpha / (-1.0 - alpha)) + ((2.0 / (-1.0 - alpha)) + ((alpha + 1.0) / pow((-1.0 - alpha), 2.0)))) / beta), (-1.0 / (-1.0 - alpha))) * (beta + (alpha + 2.0)))) / (1.0 + (2.0 + (beta + alpha)));
	}
	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)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.744835924247743e+111)
		tmp = Float64(Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / t_0) * Float64(1.0 / Float64(t_0 * Float64(beta + Float64(alpha + 3.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-1.0, Float64(Float64(Float64(alpha / Float64(-1.0 - alpha)) + Float64(Float64(2.0 / Float64(-1.0 - alpha)) + Float64(Float64(alpha + 1.0) / (Float64(-1.0 - alpha) ^ 2.0)))) / beta), Float64(-1.0 / Float64(-1.0 - alpha))) * Float64(beta + Float64(alpha + 2.0)))) / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.744835924247743e+111], N[(N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-1.0 * N[(N[(N[(alpha / N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + 1.0), $MachinePrecision] / N[Power[N[(-1.0 - alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] + N[(-1.0 / N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $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}
t_0 := \alpha + \left(\beta + 2\right)\\
\mathbf{if}\;\beta \leq 2.744835924247743 \cdot 10^{+111}:\\
\;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t_0} \cdot \frac{1}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\

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


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.7448359242477431e111

    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

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

    if 2.7448359242477431e111 < beta

    1. Initial program 9.4

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

    2. Applied egg-rr9.4

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

    3. Applied egg-rr9.4

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

    4. Taylor expanded in beta around -inf 0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.744835924247743 \cdot 10^{+111}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-1, \frac{\frac{\alpha}{-1 - \alpha} + \left(\frac{2}{-1 - \alpha} + \frac{\alpha + 1}{{\left(-1 - \alpha\right)}^{2}}\right)}{\beta}, \frac{-1}{-1 - \alpha}\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))