Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 9.9s

Alternatives: 18

Speedup: 1.7×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7000000000000:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\mathsf{fma}\left(-1, \frac{2}{-1 - \beta} + \frac{\beta}{-1 - \beta}, \frac{-1 - \beta}{{\left(-1 - \beta\right)}^{2}}\right)}{\alpha} - {\left(-1 - \beta\right)}^{-1}\right)}^{-1}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7000000000000.0)
   (/
    (*
     (+ (fma beta alpha (+ beta alpha)) 1.0)
     (pow (+ (+ beta alpha) 2.0) -2.0))
    (+ 3.0 (+ beta alpha)))
   (/
    (/
     (pow
      (-
       (/
        (fma
         -1.0
         (+ (/ 2.0 (- -1.0 beta)) (/ beta (- -1.0 beta)))
         (/ (- -1.0 beta) (pow (- -1.0 beta) 2.0)))
        alpha)
       (pow (- -1.0 beta) -1.0))
      -1.0)
     (+ (+ alpha beta) 2.0))
    (+ (+ (+ alpha beta) 1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7000000000000.0) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) * pow(((beta + alpha) + 2.0), -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = (pow(((fma(-1.0, ((2.0 / (-1.0 - beta)) + (beta / (-1.0 - beta))), ((-1.0 - beta) / pow((-1.0 - beta), 2.0))) / alpha) - pow((-1.0 - beta), -1.0)), -1.0) / ((alpha + beta) + 2.0)) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7000000000000.0)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) * (Float64(Float64(beta + alpha) + 2.0) ^ -2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64((Float64(Float64(fma(-1.0, Float64(Float64(2.0 / Float64(-1.0 - beta)) + Float64(beta / Float64(-1.0 - beta))), Float64(Float64(-1.0 - beta) / (Float64(-1.0 - beta) ^ 2.0))) / alpha) - (Float64(-1.0 - beta) ^ -1.0)) ^ -1.0) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 7000000000000.0], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(-1.0 * N[(N[(2.0 / N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - beta), $MachinePrecision] / N[Power[N[(-1.0 - beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[Power[N[(-1.0 - beta), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7e12

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   if 7e12 < alpha

   1. Initial program 76.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 76.3

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

      8. lift-*.f64 N/A

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

      9. metadata-eval 76.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 76.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\alpha \cdot \left(\frac{1}{1 + \beta} - \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)\right) + \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 55.6%

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{1 + \beta} - \left(\frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right), \alpha, \frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate--l+ N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 55.6

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

   9. Applied rewrites 55.6%

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

   10. Taylor expanded in alpha around -inf

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{-1 \cdot \frac{\left(2 \cdot \frac{1}{-1 \cdot \beta - 1} + \frac{\beta}{-1 \cdot \beta - 1}\right) - -1 \cdot \frac{1 + \beta}{{\left(-1 \cdot \beta - 1\right)}^{2}}}{\alpha} - \frac{1}{-1 \cdot \beta - 1}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 1\right) + 2} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

   12. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7000000000000:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\mathsf{fma}\left(-1, \frac{2}{-1 - \beta} + \frac{\beta}{-1 - \beta}, \frac{-1 - \beta}{{\left(-1 - \beta\right)}^{2}}\right)}{\alpha} - {\left(-1 - \beta\right)}^{-1}\right)}^{-1}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\alpha - 1}\right)}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 2.55e+90)
     (/
      (fma (+ alpha 1.0) beta (/ (fma alpha alpha -1.0) (- alpha 1.0)))
      (* (* (+ 3.0 (+ alpha beta)) t_0) t_0))
     (/
      (/
       (-
        (+ (+ 1.0 (+ alpha (pow beta -1.0))) (/ alpha beta))
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      (+ 3.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.55e+90) {
		tmp = fma((alpha + 1.0), beta, (fma(alpha, alpha, -1.0) / (alpha - 1.0))) / (((3.0 + (alpha + beta)) * t_0) * t_0);
	} else {
		tmp = ((((1.0 + (alpha + pow(beta, -1.0))) + (alpha / beta)) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.55e+90)
		tmp = Float64(fma(Float64(alpha + 1.0), beta, Float64(fma(alpha, alpha, -1.0) / Float64(alpha - 1.0))) / Float64(Float64(Float64(3.0 + Float64(alpha + beta)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(alpha + (beta ^ -1.0))) + Float64(alpha / beta)) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.55e+90], N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(N[(alpha * alpha + -1.0), $MachinePrecision] / N[(alpha - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + N[(alpha + N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5499999999999998e90

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 96.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if 2.5499999999999998e90 < beta

   1. Initial program 75.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}}{3 + \left(\beta + \alpha\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \color{blue}{\left(\left(\alpha + \frac{1}{\beta}\right) + \frac{\alpha}{\beta}\right)}\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right)} - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right)} - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right)} + \frac{\alpha}{\beta}\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(1 + \color{blue}{\left(\alpha + \frac{1}{\beta}\right)}\right) + \frac{\alpha}{\beta}\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \color{blue}{\frac{1}{\beta}}\right)\right) + \frac{\alpha}{\beta}\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \color{blue}{\frac{\alpha}{\beta}}\right) - \frac{\left(1 + \alpha\right) \cdot \left(4 + 2 \cdot \alpha\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \color{blue}{\left(1 + \alpha\right) \cdot \frac{4 + 2 \cdot \alpha}{\beta}}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \color{blue}{\left(1 + \alpha\right) \cdot \frac{4 + 2 \cdot \alpha}{\beta}}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \color{blue}{\left(1 + \alpha\right)} \cdot \frac{4 + 2 \cdot \alpha}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \color{blue}{\frac{4 + 2 \cdot \alpha}{\beta}}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{\color{blue}{2 \cdot \alpha + 4}}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

      15. lower-fma.f64 89.9

          \[\leadsto \frac{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \alpha, 4\right)}}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)} \]

   7. Applied rewrites 89.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\alpha - 1}\right)}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} - \frac{-1}{\beta}\right)\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7e+49)
   (/
    (*
     (+ (fma beta alpha (+ beta alpha)) 1.0)
     (pow (+ (+ beta alpha) 2.0) -2.0))
    (+ 3.0 (+ beta alpha)))
   (/
    (/
     (-
      (+ (+ 1.0 alpha) (- (/ alpha beta) (/ -1.0 beta)))
      (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
     (+ (+ alpha beta) 2.0))
    (+ (+ (+ alpha beta) 1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7e+49) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) * pow(((beta + alpha) + 2.0), -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((((1.0 + alpha) + ((alpha / beta) - (-1.0 / beta))) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / ((alpha + beta) + 2.0)) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7e+49)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) * (Float64(Float64(beta + alpha) + 2.0) ^ -2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + alpha) + Float64(Float64(alpha / beta) - Float64(-1.0 / beta))) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 7e+49], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(N[(alpha / beta), $MachinePrecision] - N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.9999999999999995e49

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   if 6.9999999999999995e49 < alpha

   1. Initial program 74.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 74.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 74.0

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

      8. lift-*.f64 N/A

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

      9. metadata-eval 74.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 74.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\alpha \cdot \left(\frac{1}{1 + \beta} - \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)\right) + \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 60.9%

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{1 + \beta} - \left(\frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right), \alpha, \frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate--l+ N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 60.9

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

   9. Applied rewrites 60.9%

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

   10. Taylor expanded in beta around inf

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 1\right) + 2} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-+.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-+.f64 22.8

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

   12. Applied rewrites 22.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} - \frac{-1}{\beta}\right)\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\alpha - 1}\right)}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} - \frac{-1}{\beta}\right)\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 2.55e+90)
     (/
      (fma (+ alpha 1.0) beta (/ (fma alpha alpha -1.0) (- alpha 1.0)))
      (* (* (+ 3.0 (+ alpha beta)) t_0) t_0))
     (/
      (/
       (-
        (+ (+ 1.0 alpha) (- (/ alpha beta) (/ -1.0 beta)))
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       (+ (+ alpha beta) 2.0))
      (+ (+ (+ alpha beta) 1.0) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.55e+90) {
		tmp = fma((alpha + 1.0), beta, (fma(alpha, alpha, -1.0) / (alpha - 1.0))) / (((3.0 + (alpha + beta)) * t_0) * t_0);
	} else {
		tmp = ((((1.0 + alpha) + ((alpha / beta) - (-1.0 / beta))) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / ((alpha + beta) + 2.0)) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.55e+90)
		tmp = Float64(fma(Float64(alpha + 1.0), beta, Float64(fma(alpha, alpha, -1.0) / Float64(alpha - 1.0))) / Float64(Float64(Float64(3.0 + Float64(alpha + beta)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + alpha) + Float64(Float64(alpha / beta) - Float64(-1.0 / beta))) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.55e+90], N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(N[(alpha * alpha + -1.0), $MachinePrecision] / N[(alpha - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(N[(alpha / beta), $MachinePrecision] - N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5499999999999998e90

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 96.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if 2.5499999999999998e90 < beta

   1. Initial program 75.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 75.8

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

      8. lift-*.f64 N/A

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

      9. metadata-eval 75.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 75.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 75.8

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\alpha \cdot \left(\frac{1}{1 + \beta} - \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)\right) + \left(2 \cdot \frac{1}{1 + \beta} + \frac{\beta}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 85.8%

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{1 + \beta} - \left(\frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right), \alpha, \frac{\beta}{1 + \beta} + \frac{2}{1 + \beta}\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate--l+ N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 85.8

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

   9. Applied rewrites 85.8%

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

   10. Taylor expanded in beta around inf

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 1\right) + 2} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-+.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-+.f64 90.0

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

   12. Applied rewrites 90.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\alpha - 1}\right)}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} - \frac{-1}{\beta}\right)\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\alpha - 1}\right)}{\left(t\_0 \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))) (t_1 (+ 2.0 (+ alpha beta))))
   (if (<= beta 4.4e+94)
     (/
      (fma (+ alpha 1.0) beta (/ (fma alpha alpha -1.0) (- alpha 1.0)))
      (* (* t_0 t_1) t_1))
     (/ (/ (+ alpha 1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.4e+94) {
		tmp = fma((alpha + 1.0), beta, (fma(alpha, alpha, -1.0) / (alpha - 1.0))) / ((t_0 * t_1) * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	t_1 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 4.4e+94)
		tmp = Float64(fma(Float64(alpha + 1.0), beta, Float64(fma(alpha, alpha, -1.0) / Float64(alpha - 1.0))) / Float64(Float64(t_0 * t_1) * t_1));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.4e+94], N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(N[(alpha * alpha + -1.0), $MachinePrecision] / N[(alpha - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.40000000000000024e94

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 96.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if 4.40000000000000024e94 < beta

   1. Initial program 75.8%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, \alpha, \beta\right) + 5, \beta, \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 4.7e+94)
     (/
      (fma (+ alpha 1.0) beta (+ alpha 1.0))
      (*
       (fma (+ (fma 2.0 alpha beta) 5.0) beta (* (+ 2.0 alpha) (+ 3.0 alpha)))
       t_0))
     (/ (/ (+ alpha 1.0) t_0) (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 4.7e+94) {
		tmp = fma((alpha + 1.0), beta, (alpha + 1.0)) / (fma((fma(2.0, alpha, beta) + 5.0), beta, ((2.0 + alpha) * (3.0 + alpha))) * t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 4.7e+94)
		tmp = Float64(fma(Float64(alpha + 1.0), beta, Float64(alpha + 1.0)) / Float64(fma(Float64(fma(2.0, alpha, beta) + 5.0), beta, Float64(Float64(2.0 + alpha) * Float64(3.0 + alpha))) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.7e+94], N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(2.0 * alpha + beta), $MachinePrecision] + 5.0), $MachinePrecision] * beta + N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000017e94

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in beta around 0

      \[\leadsto \frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{\color{blue}{\left(\beta \cdot \left(5 + \left(\beta + 2 \cdot \alpha\right)\right) + \left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)} \cdot \left(2 + \left(\alpha + \beta\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 96.3

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

   9. Applied rewrites 96.3%

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

   if 4.70000000000000017e94 < beta

   1. Initial program 75.8%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 4.7e+94)
     (/
      (+ (fma beta alpha (+ beta alpha)) 1.0)
      (* (* (+ 3.0 (+ beta alpha)) t_0) t_0))
     (/ (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta))) (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4.7e+94) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 4.7e+94)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(Float64(3.0 + Float64(beta + alpha)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4.7e+94], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000017e94

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.3%

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

   if 4.70000000000000017e94 < beta

   1. Initial program 75.8%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(t\_0 \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))) (t_1 (+ 2.0 (+ alpha beta))))
   (if (<= beta 2.05e+17)
     (/ (+ 1.0 beta) (* (* t_0 t_1) t_1))
     (/ (/ (+ alpha 1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.05e+17) {
		tmp = (1.0 + beta) / ((t_0 * t_1) * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_1) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    t_1 = 2.0d0 + (alpha + beta)
    if (beta <= 2.05d+17) then
        tmp = (1.0d0 + beta) / ((t_0 * t_1) * t_1)
    else
        tmp = ((alpha + 1.0d0) / t_1) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.05e+17) {
		tmp = (1.0 + beta) / ((t_0 * t_1) * t_1);
	} else {
		tmp = ((alpha + 1.0) / t_1) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	t_1 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 2.05e+17:
		tmp = (1.0 + beta) / ((t_0 * t_1) * t_1)
	else:
		tmp = ((alpha + 1.0) / t_1) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	t_1 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.05e+17)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(t_0 * t_1) * t_1));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_1) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	t_1 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 2.05e+17)
		tmp = (1.0 + beta) / ((t_0 * t_1) * t_1);
	else
		tmp = ((alpha + 1.0) / t_1) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.05e+17], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.05e17

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 95.9%

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

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 2.05e17 < beta

   1. Initial program 81.2%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{\alpha + 1}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6e+91)
   (/ (+ alpha 1.0) (* (+ 3.0 (+ alpha beta)) (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e+91) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d+91) then
        tmp = (alpha + 1.0d0) / ((3.0d0 + (alpha + beta)) * (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e+91) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6e+91:
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6e+91)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6e+91)
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6e+91], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.60000000000000034e91

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 22.7

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

   5. Applied rewrites 22.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   if 6.60000000000000034e91 < beta

   1. Initial program 75.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\alpha + 1}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+90)
   (/ (+ alpha 1.0) (* (+ 3.0 (+ alpha beta)) (+ 2.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+90) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+90) then
        tmp = (alpha + 1.0d0) / ((3.0d0 + (alpha + beta)) * (2.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+90) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+90:
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+90)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+90)
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+90], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999993e90

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 22.7

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

   5. Applied rewrites 22.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\alpha + 1}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-+.f64 21.9

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

   10. Applied rewrites 21.9%

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

   if 1.99999999999999993e90 < beta

   1. Initial program 75.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (/ (/ (+ alpha 1.0) (+ 2.0 (+ alpha beta))) (+ 3.0 (+ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha + 1.0) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) / (2.0d0 + (alpha + beta))) / (3.0d0 + (alpha + beta))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((alpha + 1.0) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((alpha + 1.0) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + Float64(alpha + beta)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around -inf

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-in N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 39.9

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

5. Applied rewrites 39.9%

   \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
6. Step-by-step derivation

7. Applied rewrites 39.9%

   \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}} \]
8. Add Preprocessing

Alternative 12: 62.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{\alpha + 1}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6e+91)
   (/ (+ alpha 1.0) (* (+ 3.0 (+ alpha beta)) (+ 2.0 beta)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e+91) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d+91) then
        tmp = (alpha + 1.0d0) / ((3.0d0 + (alpha + beta)) * (2.0d0 + beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6e+91) {
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6e+91:
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6e+91)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(3.0 + Float64(alpha + beta)) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6e+91)
		tmp = (alpha + 1.0) / ((3.0 + (alpha + beta)) * (2.0 + beta));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6e+91], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.60000000000000034e91

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 22.7

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

   5. Applied rewrites 22.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\alpha + 1}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-+.f64 21.9

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

   10. Applied rewrites 21.9%

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

   if 6.60000000000000034e91 < beta

   1. Initial program 75.8%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 88.5

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.1%

      \[\leadsto \frac{\frac{\alpha + 1}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\alpha + 1}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+90)
   (/ (+ alpha 1.0) (* (+ 2.0 beta) (+ 3.0 beta)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+90) {
		tmp = (alpha + 1.0) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+90) then
        tmp = (alpha + 1.0d0) / ((2.0d0 + beta) * (3.0d0 + beta))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+90) {
		tmp = (alpha + 1.0) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+90:
		tmp = (alpha + 1.0) / ((2.0 + beta) * (3.0 + beta))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+90)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(2.0 + beta) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+90)
		tmp = (alpha + 1.0) / ((2.0 + beta) * (3.0 + beta));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+90], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999993e90

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 22.7

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

   5. Applied rewrites 22.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\alpha + 1}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 21.6

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

   10. Applied rewrites 21.6%

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

   if 1.99999999999999993e90 < beta

   1. Initial program 75.8%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 88.5

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.1%

      \[\leadsto \frac{\frac{\alpha + 1}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+154) (/ (+ 1.0 alpha) (* beta beta)) (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000004e154

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 17.8

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 5.00000000000000004e154 < beta

   1. Initial program 68.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 87.0

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.0%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 89.0%

       \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 55.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\frac{\alpha + 1}{\beta}}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (/ (+ alpha 1.0) beta) beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) / beta) / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((alpha + 1.0) / beta) / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(alpha + 1.0) / beta) / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / beta) / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 31.3

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
6. Step-by-step derivation

7. Applied rewrites 31.7%

   \[\leadsto \frac{\frac{\alpha + 1}{\beta}}{\color{blue}{\beta}} \]
8. Add Preprocessing

Alternative 16: 52.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1 + \alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (+ 1.0 alpha) (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + alpha) / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 + alpha) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 + alpha) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 31.3

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
6. Add Preprocessing

Alternative 17: 50.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 1.0 (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0 / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 1.0 / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 1.0 / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 31.3

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
7. Step-by-step derivation

8. Applied rewrites 30.3%

   \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
9. Add Preprocessing

Alternative 18: 32.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ alpha (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = alpha / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return alpha / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(alpha / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = alpha / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 31.3

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around inf

   \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 20.1%

   \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(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)))