Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.8%

Time: 10.8s

Alternatives: 17

Speedup: 2.0×

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: 94.5% 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, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\frac{{t\_1}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(1 + \alpha\right)}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

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 2e+132)
     (/ (* (pow t_1 -2.0) (+ 1.0 (fma beta alpha (+ alpha beta)))) t_0)
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (+ 2.0 alpha) beta) (+ 1.0 alpha)))
       t_0)
      t_1))))

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 <= 2e+132) {
		tmp = (pow(t_1, -2.0) * (1.0 + fma(beta, alpha, (alpha + beta)))) / t_0;
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - (((2.0 + alpha) / beta) * (1.0 + alpha))) / t_0) / t_1;
	}
	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 <= 2e+132)
		tmp = Float64(Float64((t_1 ^ -2.0) * Float64(1.0 + fma(beta, alpha, Float64(alpha + beta)))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(Float64(2.0 + alpha) / beta) * Float64(1.0 + alpha))) / t_0) / t_1);
	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, 2e+132], N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999998e132

   1. Initial program 99.3%

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

   4. Applied rewrites 99.4%

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

   if 1.99999999999999998e132 < beta

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\frac{{\left(2 + \left(\alpha + \beta\right)\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)\right)}{3 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(1 + \alpha\right)}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_1}}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(1 + \alpha\right)}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

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 3.2e+132)
     (/ (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_1) (* t_0 t_1))
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (+ 2.0 alpha) beta) (+ 1.0 alpha)))
       t_0)
      t_1))))

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 <= 3.2e+132) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / t_1) / (t_0 * t_1);
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - (((2.0 + alpha) / beta) * (1.0 + alpha))) / t_0) / t_1;
	}
	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 <= 3.2e+132)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_1) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(Float64(2.0 + alpha) / beta) * Float64(1.0 + alpha))) / t_0) / t_1);
	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, 3.2e+132], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1999999999999997e132

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      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. lower-/.f64 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. Applied rewrites 99.4%

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

   if 3.1999999999999997e132 < beta

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(1 + \alpha\right)}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.2e+132)
     (/
      (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
      (* (+ 3.0 (+ alpha beta)) t_0))
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (fma 2.0 alpha 5.0) beta) (+ 1.0 alpha)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.2e+132) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / ((3.0 + (alpha + beta)) * t_0);
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - ((fma(2.0, alpha, 5.0) / beta) * (1.0 + alpha))) / beta) / t_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 <= 3.2e+132)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(Float64(3.0 + Float64(alpha + beta)) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(fma(2.0, alpha, 5.0) / beta) * Float64(1.0 + alpha))) / beta) / 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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.2e+132], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * alpha + 5.0), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1999999999999997e132

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      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. lower-/.f64 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. Applied rewrites 99.4%

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

   if 3.1999999999999997e132 < beta

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-3, \frac{\alpha}{\beta}, 1 - \frac{9}{\beta}\right), \alpha, \frac{\beta - 6}{\beta}\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.2e+132)
     (/
      (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
      (* (+ 3.0 (+ alpha beta)) t_0))
     (/
      (/
       (fma
        (fma -3.0 (/ alpha beta) (- 1.0 (/ 9.0 beta)))
        alpha
        (/ (- beta 6.0) beta))
       beta)
      beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.2e+132) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / ((3.0 + (alpha + beta)) * t_0);
	} else {
		tmp = (fma(fma(-3.0, (alpha / beta), (1.0 - (9.0 / beta))), alpha, ((beta - 6.0) / beta)) / beta) / 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 <= 3.2e+132)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(Float64(3.0 + Float64(alpha + beta)) * t_0));
	else
		tmp = Float64(Float64(fma(fma(-3.0, Float64(alpha / beta), Float64(1.0 - Float64(9.0 / beta))), alpha, Float64(Float64(beta - 6.0) / beta)) / beta) / 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, 3.2e+132], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-3.0 * N[(alpha / beta), $MachinePrecision] + N[(1.0 - N[(9.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * alpha + N[(N[(beta - 6.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1999999999999997e132

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      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. lower-/.f64 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. Applied rewrites 99.4%

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

   if 3.1999999999999997e132 < beta

   1. Initial program 83.3%

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

   3. Taylor expanded in beta around -inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 87.6%

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

   10. Applied rewrites 87.0%

       \[\leadsto \frac{\frac{1 - \frac{6}{\beta}}{\beta}}{\color{blue}{\beta}} \]

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{\left(1 + \alpha \cdot \left(\left(1 + -3 \cdot \frac{\alpha}{\beta}\right) - 9 \cdot \frac{1}{\beta}\right)\right) - 6 \cdot \frac{1}{\beta}}{\beta}}{\beta} \]
   12. Step-by-step derivation

   13. Applied rewrites 91.8%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-3, \frac{\alpha}{\beta}, 1 - \frac{9}{\beta}\right), \alpha, \frac{\beta - 6}{\beta}\right)}{\beta}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-3, \frac{\alpha}{\beta}, 1 - \frac{9}{\beta}\right), \alpha, \frac{\beta - 6}{\beta}\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 2.7e+132)
     (/
      (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
      (* (+ 3.0 (+ alpha beta)) t_0))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.7e+132) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / ((3.0 + (alpha + beta)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	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 <= 2.7e+132)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(Float64(3.0 + Float64(alpha + beta)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	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, 2.7e+132], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e132

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      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. lower-/.f64 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. Applied rewrites 99.4%

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

   if 2.7e132 < beta

   1. Initial program 83.3%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(t\_1 \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))) (t_1 (+ 3.0 (+ alpha beta))))
   (if (<= beta 2.4e+92)
     (/ (+ 1.0 (fma beta alpha (+ alpha beta))) (* (* t_1 t_0) t_0))
     (/ (/ (+ 1.0 alpha) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.4e+92) {
		tmp = (1.0 + fma(beta, alpha, (alpha + beta))) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.4e+92)
		tmp = Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / Float64(Float64(t_1 * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.4e+92], N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.40000000000000005e92

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      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 96.1%

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

   if 2.40000000000000005e92 < beta

   1. Initial program 86.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 93.4

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

   7. Applied rewrites 93.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 1.22e+16)
     (/ (+ 1.0 beta) (* (* (+ 2.0 beta) (+ 3.0 beta)) t_0))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.22e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	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) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 1.22d+16) then
        tmp = (1.0d0 + beta) / (((2.0d0 + beta) * (3.0d0 + beta)) * t_0)
    else
        tmp = ((1.0d0 + alpha) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.22e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 1.22e+16:
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0)
	else:
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0)
	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 <= 1.22e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(Float64(2.0 + beta) * Float64(3.0 + beta)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 1.22e+16)
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	else
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.22e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.22e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      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%

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

   5. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 1.22e16 < beta

   1. Initial program 88.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 1.22e+16)
     (/ (+ 1.0 beta) (* (* (+ 2.0 beta) (+ 3.0 beta)) t_0))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.22e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / 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) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 1.22d+16) then
        tmp = (1.0d0 + beta) / (((2.0d0 + beta) * (3.0d0 + beta)) * t_0)
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (alpha + beta))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.22e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 1.22e+16:
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0)
	else:
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_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 <= 1.22e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(Float64(2.0 + beta) * Float64(3.0 + beta)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(alpha + beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 1.22e+16)
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * t_0);
	else
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / 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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.22e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.22e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      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%

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

   5. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 1.22e16 < beta

   1. Initial program 88.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+16)
   (/ (+ 1.0 beta) (* (* (+ 2.0 beta) (+ 3.0 beta)) (+ 2.0 (+ alpha beta))))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + 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+16) then
        tmp = (1.0d0 + beta) / (((2.0d0 + beta) * (3.0d0 + beta)) * (2.0d0 + (alpha + beta)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+16) {
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * (2.0 + (alpha + beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+16:
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * (2.0 + (alpha + beta)))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(Float64(2.0 + beta) * Float64(3.0 + beta)) * Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+16)
		tmp = (1.0 + beta) / (((2.0 + beta) * (3.0 + beta)) * (2.0 + (alpha + beta)));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + 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+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      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%

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

   5. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 2e16 < beta

   1. Initial program 88.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 91.1

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

   5. Applied rewrites 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 91.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 91.1

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

   7. Applied rewrites 91.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(2 + \beta\right) \cdot \left(3 + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{0.5}{t\_0}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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))))
   (if (<= beta 6.0)
     (/ (/ 0.5 t_0) (+ 2.0 (+ alpha beta)))
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.0) {
		tmp = (0.5 / t_0) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / 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) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    if (beta <= 6.0d0) then
        tmp = (0.5d0 / t_0) / (2.0d0 + (alpha + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / 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 tmp;
	if (beta <= 6.0) {
		tmp = (0.5 / t_0) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 6.0:
		tmp = (0.5 / t_0) / (2.0 + (alpha + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(0.5 / t_0) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / 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);
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = (0.5 / t_0) / (2.0 + (alpha + beta));
	else
		tmp = ((1.0 + alpha) / beta) / 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]}, If[LessEqual[beta, 6.0], N[(N[(0.5 / t$95$0), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 82.7%

       \[\leadsto \frac{\frac{0.5}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   if 6 < beta

   1. Initial program 89.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.7

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

   5. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 88.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{0.5}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((1.0 + 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 = ((1.0d0 + alpha) / beta) / (3.0d0 + (alpha + beta))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((1.0 + 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[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 30.4

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

5. Applied rewrites 30.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 30.4

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 30.4

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

7. Applied rewrites 30.4%

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

8. Final simplification 30.4%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)} \]
9. Add Preprocessing

Alternative 12: 55.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.4e+154)
   (/ (+ 1.0 alpha) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4e+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 <= 1.4d+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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, 1.4e+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 < 1.4e154

   1. Initial program 99.3%

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

   3. 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 18.1

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

   5. Applied rewrites 18.1%

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

   if 1.4e154 < beta

   1. Initial program 82.1%

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

   3. 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 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 86.8%

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

   10. Applied rewrites 90.5%

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

Alternative 13: 56.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) / (3.0 + 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) / (3.0d0 + beta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 30.4

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

5. Applied rewrites 30.4%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
7. Step-by-step derivation

   1. lower-+.f64 30.3

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]

8. Applied rewrites 30.3%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
9. Add Preprocessing

Alternative 14: 55.9% accurate, 3.2× speedup

Math

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

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(Float64(1.0 + alpha) / 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[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. 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 29.9

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

5. Applied rewrites 29.9%

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

7. Applied rewrites 30.7%

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

Alternative 15: 53.0% 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 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. 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 29.9

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

5. Applied rewrites 29.9%

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

Alternative 16: 49.8% 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 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. 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 29.9

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

5. Applied rewrites 29.9%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 28.8%

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

Alternative 17: 31.8% 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 96.4%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. 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 29.9

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

5. Applied rewrites 29.9%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 18.5%

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

Reproduce

herbie shell --seed 2024268 
(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)))