Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.6%

Time: 14.8s

Alternatives: 22

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+119}:\\ \;\;\;\;{\left(\alpha + \left(\beta + 2\right)\right)}^{-2} \cdot \frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+119)
   (*
    (pow (+ alpha (+ beta 2.0)) -2.0)
    (/ (+ (fma alpha beta (+ beta alpha)) 1.0) (+ alpha (+ beta 3.0))))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+119) {
		tmp = pow((alpha + (beta + 2.0)), -2.0) * ((fma(alpha, beta, (beta + alpha)) + 1.0) / (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+119)
		tmp = Float64((Float64(alpha + Float64(beta + 2.0)) ^ -2.0) * Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1e+119], N[(N[Power[N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999944e118

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. pow-prod-up N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. metadata-eval 97.3

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

   6. Applied rewrites 97.3%

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

   if 9.99999999999999944e118 < beta

   1. Initial program 68.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 82.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+119}:\\ \;\;\;\;{\left(\alpha + \left(\beta + 2\right)\right)}^{-2} \cdot \frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0} \cdot \frac{\frac{\left(1 - \alpha \cdot \alpha\right) \cdot \left(\beta + 1\right)}{1 - \alpha}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 5e+107)
     (*
      (/ (/ 1.0 t_0) t_0)
      (/
       (/ (* (- 1.0 (* alpha alpha)) (+ beta 1.0)) (- 1.0 alpha))
       (+ alpha (+ beta 3.0))))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+107) {
		tmp = ((1.0 / t_0) / t_0) * ((((1.0 - (alpha * alpha)) * (beta + 1.0)) / (1.0 - alpha)) / (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 5d+107) then
        tmp = ((1.0d0 / t_0) / t_0) * ((((1.0d0 - (alpha * alpha)) * (beta + 1.0d0)) / (1.0d0 - alpha)) / (alpha + (beta + 3.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+107) {
		tmp = ((1.0 / t_0) / t_0) * ((((1.0 - (alpha * alpha)) * (beta + 1.0)) / (1.0 - alpha)) / (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5e+107:
		tmp = ((1.0 / t_0) / t_0) * ((((1.0 - (alpha * alpha)) * (beta + 1.0)) / (1.0 - alpha)) / (alpha + (beta + 3.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5e+107)
		tmp = Float64(Float64(Float64(1.0 / t_0) / t_0) * Float64(Float64(Float64(Float64(1.0 - Float64(alpha * alpha)) * Float64(beta + 1.0)) / Float64(1.0 - alpha)) / Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 5e+107)
		tmp = ((1.0 / t_0) / t_0) * ((((1.0 - (alpha * alpha)) * (beta + 1.0)) / (1.0 - alpha)) / (alpha + (beta + 3.0)));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+107], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(N[(1.0 - N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000002e107

   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. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. flip-+ N/A

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

      19. associate-*l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval N/A

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

      23. lower--.f64 N/A

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

      24. lower-*.f64 N/A

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

      25. lift-+.f64 N/A

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

      26. +-commutative N/A

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

      27. lower-+.f64 N/A

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

      28. lower--.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if 5.0000000000000002e107 < beta

   1. Initial program 67.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 80.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{\left(1 - \alpha \cdot \alpha\right) \cdot \left(\beta + 1\right)}{1 - \alpha}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2e+111)
     (*
      (/ (+ (fma alpha beta (+ beta alpha)) 1.0) (+ alpha (+ beta 3.0)))
      (/ (/ 1.0 t_0) t_0))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2e+111) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / (alpha + (beta + 3.0))) * ((1.0 / t_0) / t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2e+111)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 / t_0) / t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+111], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999991e111

   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. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.7%

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

   if 1.99999999999999991e111 < beta

   1. Initial program 67.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 80.8%

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

4. Final simplification 94.7%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2.5e+111)
     (/
      (/ (+ (fma alpha beta (+ beta alpha)) 1.0) t_0)
      (* (+ alpha (+ beta 3.0)) t_0))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / t_0) / ((alpha + (beta + 3.0)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.5e+111)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+111], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4999999999999998e111

   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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 96.9%

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

   if 2.4999999999999998e111 < beta

   1. Initial program 67.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 80.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{2 + \left(\beta + \alpha\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := -2 - \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_0}}{\left(\left(\beta + \alpha\right) + 3\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (+ beta alpha))))
   (if (<= beta 2.5e+111)
     (/ (/ (* (+ beta 1.0) (+ alpha 1.0)) t_0) (* (+ (+ beta alpha) 3.0) t_0))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = -2.0 - (beta + alpha);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = (((beta + 1.0) * (alpha + 1.0)) / t_0) / (((beta + alpha) + 3.0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) - (beta + alpha)
    if (beta <= 2.5d+111) then
        tmp = (((beta + 1.0d0) * (alpha + 1.0d0)) / t_0) / (((beta + alpha) + 3.0d0) * t_0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = -2.0 - (beta + alpha);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = (((beta + 1.0) * (alpha + 1.0)) / t_0) / (((beta + alpha) + 3.0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = -2.0 - (beta + alpha)
	tmp = 0
	if beta <= 2.5e+111:
		tmp = (((beta + 1.0) * (alpha + 1.0)) / t_0) / (((beta + alpha) + 3.0) * t_0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(-2.0 - Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.5e+111)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / t_0) / Float64(Float64(Float64(beta + alpha) + 3.0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = -2.0 - (beta + alpha);
	tmp = 0.0;
	if (beta <= 2.5e+111)
		tmp = (((beta + 1.0) * (alpha + 1.0)) / t_0) / (((beta + alpha) + 3.0) * t_0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+111], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4999999999999998e111

   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 alpha around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 96.9%

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

   if 2.4999999999999998e111 < beta

   1. Initial program 67.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 80.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{-2 - \left(\beta + \alpha\right)}}{\left(\left(\beta + \alpha\right) + 3\right) \cdot \left(-2 - \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \alpha\right) + 3}}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.5e+111)
     (/ (/ (* (+ beta 1.0) (+ alpha 1.0)) (+ (+ beta alpha) 3.0)) (* t_0 t_0))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = (((beta + 1.0) * (alpha + 1.0)) / ((beta + alpha) + 3.0)) / (t_0 * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.5d+111) then
        tmp = (((beta + 1.0d0) * (alpha + 1.0d0)) / ((beta + alpha) + 3.0d0)) / (t_0 * t_0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = (((beta + 1.0) * (alpha + 1.0)) / ((beta + alpha) + 3.0)) / (t_0 * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.5e+111:
		tmp = (((beta + 1.0) * (alpha + 1.0)) / ((beta + alpha) + 3.0)) / (t_0 * t_0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.5e+111)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(Float64(beta + alpha) + 3.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.5e+111)
		tmp = (((beta + 1.0) * (alpha + 1.0)) / ((beta + alpha) + 3.0)) / (t_0 * t_0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+111], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4999999999999998e111

   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. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 90.4%

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

   6. Applied rewrites 96.9%

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

   if 2.4999999999999998e111 < beta

   1. Initial program 67.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 80.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \alpha\right) + 3}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_1 \cdot \left(t\_1 \cdot \left(\left(\beta + \alpha\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 (+ beta alpha))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 7.8e+100)
     (/
      (+ (fma alpha beta (+ beta alpha)) 1.0)
      (* t_1 (* t_1 (+ (+ beta alpha) 3.0))))
     (/ (/ (+ alpha 1.0) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / (t_1 * (t_1 * ((beta + alpha) + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 7.8e+100)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_1 * Float64(t_1 * Float64(Float64(beta + alpha) + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.8e+100], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 * N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e100

   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. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-+l+ N/A

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

      19. lower-+.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if 7.8e100 < beta

   1. Initial program 69.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0 \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 (+ beta alpha))))
   (if (<= beta 7.8e+100)
     (/
      (+ (fma alpha beta (+ beta alpha)) 1.0)
      (* t_0 (* (+ alpha (+ beta 3.0)) t_0)))
     (/ (/ (+ alpha 1.0) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.8e+100)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.8e+100], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e100

   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 91.2%

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

   if 7.8e100 < beta

   1. Initial program 69.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := -2 - \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_1 \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 (+ beta alpha))) (t_1 (- -2.0 (+ beta alpha))))
   (if (<= beta 7.8e+100)
     (/ (* (+ beta 1.0) (+ alpha 1.0)) (* t_1 (* (+ alpha (+ beta 3.0)) t_1)))
     (/ (/ (+ alpha 1.0) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = -2.0 - (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * ((alpha + (beta + 3.0)) * t_1));
	} else {
		tmp = ((alpha + 1.0) / 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    t_1 = (-2.0d0) - (beta + alpha)
    if (beta <= 7.8d+100) then
        tmp = ((beta + 1.0d0) * (alpha + 1.0d0)) / (t_1 * ((alpha + (beta + 3.0d0)) * t_1))
    else
        tmp = ((alpha + 1.0d0) / 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 + (beta + alpha);
	double t_1 = -2.0 - (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * ((alpha + (beta + 3.0)) * t_1));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = -2.0 - (beta + alpha)
	tmp = 0
	if beta <= 7.8e+100:
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * ((alpha + (beta + 3.0)) * t_1))
	else:
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(-2.0 - Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.8e+100)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(t_1 * Float64(Float64(alpha + Float64(beta + 3.0)) * t_1)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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 + (beta + alpha);
	t_1 = -2.0 - (beta + alpha);
	tmp = 0.0;
	if (beta <= 7.8e+100)
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * ((alpha + (beta + 3.0)) * t_1));
	else
		tmp = ((alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.8e+100], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e100

   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 alpha around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 91.2%

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

   if 7.8e100 < beta

   1. Initial program 69.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(-2 - \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 (+ beta alpha))))
   (if (<= beta 7.8e+100)
     (/ (* (+ beta 1.0) (+ alpha 1.0)) (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (/ (/ (+ alpha 1.0) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / 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 + (beta + alpha)
    if (beta <= 7.8d+100) then
        tmp = ((beta + 1.0d0) * (alpha + 1.0d0)) / ((alpha + (beta + 3.0d0)) * (t_0 * t_0))
    else
        tmp = ((alpha + 1.0d0) / 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 + (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 7.8e+100:
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0))
	else:
		tmp = ((alpha + 1.0) / t_0) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.8e+100)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / 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 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 7.8e+100)
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	else
		tmp = ((alpha + 1.0) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.8e+100], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e100

   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. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 91.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 91.2

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if 7.8e100 < beta

   1. Initial program 69.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

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 (+ beta alpha))))
   (if (<= beta 7.8e+100)
     (/ (* (+ beta 1.0) (+ alpha 1.0)) (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (/ (/ (+ alpha 1.0) beta) (+ (+ beta alpha) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	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 + (beta + alpha)
    if (beta <= 7.8d+100) then
        tmp = ((beta + 1.0d0) * (alpha + 1.0d0)) / ((alpha + (beta + 3.0d0)) * (t_0 * t_0))
    else
        tmp = ((alpha + 1.0d0) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.8e+100) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 7.8e+100:
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0))
	else:
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.8e+100)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 7.8e+100)
		tmp = ((beta + 1.0) * (alpha + 1.0)) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	else
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.8e+100], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8e100

   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. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 91.2

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 91.2

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if 7.8e100 < beta

   1. Initial program 69.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 79.4

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

   5. Applied rewrites 79.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

   7. Applied rewrites 79.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3e+41)
   (/
    (/ (+ beta 1.0) (fma beta (+ beta 4.0) 4.0))
    (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ beta alpha) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3e+41) {
		tmp = ((beta + 1.0) / fma(beta, (beta + 4.0), 4.0)) / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3e+41)
		tmp = Float64(Float64(Float64(beta + 1.0) / fma(beta, Float64(beta + 4.0), 4.0)) / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.3e+41], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e41

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.5

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 70.5%

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

   if 1.3e41 < beta

   1. Initial program 77.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

   7. Applied rewrites 73.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 14:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 14.0)
   (/ (+ alpha 1.0) (* (* (+ alpha 2.0) (+ alpha 2.0)) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ beta alpha) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.0) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	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) :: tmp
    if (beta <= 14.0d0) then
        tmp = (alpha + 1.0d0) / (((alpha + 2.0d0) * (alpha + 2.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 14.0) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 14.0:
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 14.0)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 14.0)
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	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, 14.0], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 14

   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. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

   if 14 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

   7. Applied rewrites 72.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 14:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta \cdot \beta, -0.0625, 0.25\right)}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.75)
   (/ (fma (* beta beta) -0.0625 0.25) (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ beta alpha) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75) {
		tmp = fma((beta * beta), -0.0625, 0.25) / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.75)
		tmp = Float64(fma(Float64(beta * beta), -0.0625, 0.25) / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.75], N[(N[(N[(beta * beta), $MachinePrecision] * -0.0625 + 0.25), $MachinePrecision] / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{-1}{16} \cdot {\beta}^{2}}}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.7%

       \[\leadsto \frac{\mathsf{fma}\left(\beta \cdot \beta, \color{blue}{-0.0625}, 0.25\right)}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 1.75 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\beta}}{\color{blue}{\left(\alpha + \beta\right) + 3}} \]

   7. Applied rewrites 72.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta \cdot \beta, -0.0625, 0.25\right)}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta \cdot \beta, -0.0625, 0.25\right)}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.75)
   (/ (fma (* beta beta) -0.0625 0.25) (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75) {
		tmp = fma((beta * beta), -0.0625, 0.25) / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.75)
		tmp = Float64(fma(Float64(beta * beta), -0.0625, 0.25) / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.75], N[(N[(N[(beta * beta), $MachinePrecision] * -0.0625 + 0.25), $MachinePrecision] / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{-1}{16} \cdot {\beta}^{2}}}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.7%

       \[\leadsto \frac{\mathsf{fma}\left(\beta \cdot \beta, \color{blue}{-0.0625}, 0.25\right)}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 1.75 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 72.5

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

   8. Applied rewrites 72.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta \cdot \beta, -0.0625, 0.25\right)}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.25 (+ 2.0 (+ (+ beta alpha) 1.0)))
   (if (<= beta 1.35e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (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 <= 6.6d0) then
        tmp = 0.25d0 / (2.0d0 + ((beta + alpha) + 1.0d0))
    else if (beta <= 1.35d+154) then
        tmp = (alpha + 1.0d0) / (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 <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0))
	elif beta <= 1.35e+154:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.25 / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(alpha + 1.0) / 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 <= 6.6)
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	elseif (beta <= 1.35e+154)
		tmp = (alpha + 1.0) / (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, 6.6], N[(0.25 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.5999999999999996

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 70.6%

       \[\leadsto \frac{0.25}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 6.5999999999999996 < beta < 1.35000000000000003e154

   1. Initial program 80.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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 54.2

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

   5. Applied rewrites 54.2%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 76.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 91.0%

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

   10. Applied rewrites 91.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 0.25 (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	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) :: tmp
    if (beta <= 4.5d0) then
        tmp = 0.25d0 / (2.0d0 + ((beta + alpha) + 1.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(0.25 / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	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, 4.5], N[(0.25 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 70.6%

       \[\leadsto \frac{0.25}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 4.5 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 72.5

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

   8. Applied rewrites 72.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.25 (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d0) then
        tmp = 0.25d0 / (2.0d0 + ((beta + alpha) + 1.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.25 / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(0.25 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 70.6%

       \[\leadsto \frac{0.25}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 6.5999999999999996 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 72.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 94.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.25 (+ 2.0 (+ (+ beta alpha) 1.0)))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d0) then
        tmp = 0.25d0 / (2.0d0 + ((beta + alpha) + 1.0d0))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.25 / Float64(2.0 + Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.25 / (2.0 + ((beta + alpha) + 1.0));
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(0.25 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 70.6%

       \[\leadsto \frac{0.25}{\left(1 + \left(\alpha + \beta\right)\right) + 2} \]

   if 6.5999999999999996 < beta

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{2 + \left(\left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.25e-33) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.25e-33) {
		tmp = 1.0 / (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 (alpha <= 1.25d-33) then
        tmp = 1.0d0 / (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 (alpha <= 1.25e-33) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.25e-33:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.25e-33)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.25e-33)
		tmp = 1.0 / (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[alpha, 1.25e-33], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.25000000000000007e-33

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 29.3%

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

   if 1.25000000000000007e-33 < alpha

   1. Initial program 82.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 11.6

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 11.6%

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

Alternative 21: 52.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 23.3

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

5. Applied rewrites 23.3%

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

6. Final simplification 23.3%

   \[\leadsto \frac{\alpha + 1}{\beta \cdot \beta} \]
7. Add Preprocessing

Alternative 22: 31.4% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 23.3

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

5. Applied rewrites 23.3%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 15.0%

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

Reproduce

herbie shell --seed 2024225 
(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)))