Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.7%

Time: 3.8s

Alternatives: 18

Speedup: 2.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\mathsf{fma}\left(\beta, 4 + \left(\beta + \left(\alpha + \alpha\right)\right), \left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\alpha + \frac{1 + \alpha}{\beta}\right)\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\frac{3 + \alpha}{\beta} + 1\right) \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+76)
   (/
    (/
     (+ (fma beta alpha (+ beta alpha)) 1.0)
     (fma
      beta
      (+ 4.0 (+ beta (+ alpha alpha)))
      (* (+ 2.0 alpha) (+ 2.0 alpha))))
    (+ 3.0 (+ beta alpha)))
   (/
    (/
     (-
      (+ 1.0 (+ alpha (/ (+ 1.0 alpha) beta)))
      (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
     (+ (+ alpha beta) 2.0))
    (* (+ (/ (+ 3.0 alpha) beta) 1.0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+76) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / fma(beta, (4.0 + (beta + (alpha + alpha))), ((2.0 + alpha) * (2.0 + alpha)))) / (3.0 + (beta + alpha));
	} else {
		tmp = (((1.0 + (alpha + ((1.0 + alpha) / beta))) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / ((alpha + beta) + 2.0)) / ((((3.0 + alpha) / beta) + 1.0) * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+76)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / fma(beta, Float64(4.0 + Float64(beta + Float64(alpha + alpha))), Float64(Float64(2.0 + alpha) * Float64(2.0 + alpha)))) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(alpha + Float64(Float64(1.0 + alpha) / beta))) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(Float64(Float64(3.0 + alpha) / beta) + 1.0) * 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+76], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(beta * N[(4.0 + N[(beta + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(alpha + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(3.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e76

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in beta around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. pow2 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if 1e76 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. div-add N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. div-add N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 55.9

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

   9. Applied rewrites 55.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\mathsf{fma}\left(\beta, 4 + \left(\beta + \left(\alpha + \alpha\right)\right), \left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+76)
   (/
    (/
     (+ (fma beta alpha (+ beta alpha)) 1.0)
     (fma
      beta
      (+ 4.0 (+ beta (+ alpha alpha)))
      (* (+ 2.0 alpha) (+ 2.0 alpha))))
    (+ 3.0 (+ beta alpha)))
   (/
    (/
     (-
      (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
      (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
     beta)
    (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+76) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / fma(beta, (4.0 + (beta + (alpha + alpha))), ((2.0 + alpha) * (2.0 + alpha)))) / (3.0 + (beta + alpha));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+76)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / fma(beta, Float64(4.0 + Float64(beta + Float64(alpha + alpha))), Float64(Float64(2.0 + alpha) * Float64(2.0 + alpha)))) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1e+76], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(beta * N[(4.0 + N[(beta + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e76

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in beta around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. pow2 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if 1e76 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 55.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\mathsf{fma}\left(\beta, 4 + \left(\beta + \left(\alpha + \alpha\right)\right), \left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+76)
   (/
    (/
     (+ (fma beta alpha (+ beta alpha)) 1.0)
     (fma
      beta
      (+ 4.0 (+ beta (+ alpha alpha)))
      (* (+ 2.0 alpha) (+ 2.0 alpha))))
    (+ 3.0 (+ beta alpha)))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+76) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / fma(beta, (4.0 + (beta + (alpha + alpha))), ((2.0 + alpha) * (2.0 + alpha)))) / (3.0 + (beta + alpha));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+76)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / fma(beta, Float64(4.0 + Float64(beta + Float64(alpha + alpha))), Float64(Float64(2.0 + alpha) * Float64(2.0 + alpha)))) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1e+76], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(beta * N[(4.0 + N[(beta + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e76

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

   4. Taylor expanded in beta around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. pow2 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if 1e76 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0}}{t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2.3e+115)
     (/
      (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_0)
      (* t_0 (+ 3.0 (+ beta alpha))))
     (/
      (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
      (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2.3e+115) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / t_0) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2.3e+115)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_0) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.3e+115], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.30000000000000004e115

   1. Initial program 94.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-+.f64 N/A

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

   3. Applied rewrites 93.0%

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

   if 2.30000000000000004e115 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 5: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0 \cdot t\_0}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+76)
     (/
      (/ (+ (fma beta alpha (+ beta alpha)) 1.0) (* t_0 t_0))
      (+ 3.0 (+ beta alpha)))
     (/
      (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
      (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+76) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / (t_0 * t_0)) / (3.0 + (beta + alpha));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+76)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(t_0 * t_0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+76], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e76

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

   if 1e76 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 6: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0 \cdot \left(t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+86)
     (/
      (+ (fma beta alpha (+ beta alpha)) 1.0)
      (* t_0 (* t_0 (+ 3.0 (+ beta alpha)))))
     (/
      (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
      (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+86) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (t_0 * (t_0 * (3.0 + (beta + alpha))));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+86)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(t_0 * Float64(t_0 * Float64(3.0 + Float64(beta + alpha)))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+86], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e86

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.3

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

   7. Applied rewrites 84.3%

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

   if 1e86 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 7: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(t\_0 \cdot t\_0\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+86)
     (/
      (+ (fma beta alpha (+ beta alpha)) 1.0)
      (* (* t_0 t_0) (+ 3.0 (+ beta alpha))))
     (/
      (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
      (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+86) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / ((t_0 * t_0) * (3.0 + (beta + alpha)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+86)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(t_0 * t_0) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+86], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e86

   1. Initial program 94.4%

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

   3. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 84.3%

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

   if 1e86 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 8: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 225000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 225000000000.0)
   (/
    (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 2.0 beta)))
    (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 225000000000.0d0) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (2.0d0 + beta))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 225000000000.0:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 225000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(2.0 + beta))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 225000000000.0)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (2.0 + beta))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 225000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25e11

   1. Initial program 94.4%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 92.7

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

   4. Applied rewrites 92.7%

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

   if 2.25e11 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 225000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 225000000000.0)
   (/ (+ 1.0 beta) (* (+ 3.0 beta) (* (+ 2.0 beta) (+ 2.0 beta))))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 225000000000.0d0) then
        tmp = (1.0d0 + beta) / ((3.0d0 + beta) * ((2.0d0 + beta) * (2.0d0 + beta)))
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 225000000000.0:
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)))
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 225000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * Float64(Float64(2.0 + beta) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 225000000000.0)
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 225000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25e11

   1. Initial program 94.4%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 85.2

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

   4. Applied rewrites 85.2%

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

   if 2.25e11 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lift-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 10: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 225000000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 225000000000.0)
   (/ (+ 1.0 beta) (* (+ 3.0 beta) (* (+ 2.0 beta) (+ 2.0 beta))))
   (/ (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0)) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 225000000000.0d0) then
        tmp = (1.0d0 + beta) / ((3.0d0 + beta) * ((2.0d0 + beta) * (2.0d0 + beta)))
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225000000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 225000000000.0:
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)))
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 225000000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * Float64(Float64(2.0 + beta) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 225000000000.0)
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) * (2.0 + beta)));
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + 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, 225000000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25e11

   1. Initial program 94.4%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 85.2

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

   4. Applied rewrites 85.2%

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

   if 2.25e11 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 11: 97.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (/ 0.25 (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))
   (/ (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0)) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.25d0 / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0)
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = Float64(0.25 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / (3.0 + 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, 2.1], N[(0.25 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 47.2%

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

   if 2.10000000000000009 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 94.3

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

      3. lift-*.f64 N/A

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

      4. metadata-eval 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.7

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

   9. Applied rewrites 62.7%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 12: 96.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.7)
   (/ 0.25 (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))
   (if (<= beta 8.8e+157)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else if (beta <= 8.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.7d0) then
        tmp = 0.25d0 / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
    else if (beta <= 8.8d+157) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else if (beta <= 8.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.7:
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0)
	elif beta <= 8.8e+157:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.7)
		tmp = Float64(0.25 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	elseif (beta <= 8.8e+157)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.7)
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	elseif (beta <= 8.8e+157)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.7], N[(0.25 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8.8e+157], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.70000000000000018

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 47.2%

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

   if 6.70000000000000018 < beta < 8.8000000000000005e157

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   if 8.8000000000000005e157 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      5. lower-/.f64 35.1

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

   9. Applied rewrites 35.1%

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

Alternative 13: 96.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.7:\\ \;\;\;\;\frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta} + \frac{\alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.7)
   (/ 0.25 (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))
   (if (<= beta 8.8e+157)
     (+ (/ 1.0 (* beta beta)) (/ alpha (* beta beta)))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else if (beta <= 8.8e+157) {
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.7d0) then
        tmp = 0.25d0 / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
    else if (beta <= 8.8d+157) then
        tmp = (1.0d0 / (beta * beta)) + (alpha / (beta * beta))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.7) {
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	} else if (beta <= 8.8e+157) {
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.7:
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0)
	elif beta <= 8.8e+157:
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.7)
		tmp = Float64(0.25 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	elseif (beta <= 8.8e+157)
		tmp = Float64(Float64(1.0 / Float64(beta * beta)) + Float64(alpha / Float64(beta * beta)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.7)
		tmp = 0.25 / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	elseif (beta <= 8.8e+157)
		tmp = (1.0 / (beta * beta)) + (alpha / (beta * beta));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.7], N[(0.25 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8.8e+157], N[(N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.70000000000000018

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 47.2%

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

   if 6.70000000000000018 < beta < 8.8000000000000005e157

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. div-add-rev N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 8.8000000000000005e157 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      5. lower-/.f64 35.1

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

   9. Applied rewrites 35.1%

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

Alternative 14: 55.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.8e+157)
   (/ (+ 1.0 alpha) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.8d+157) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e+157) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.8e+157:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.8e+157)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.8e+157)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.8e+157], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.8000000000000005e157

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   if 8.8000000000000005e157 < beta

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      5. lower-/.f64 35.1

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

   9. Applied rewrites 35.1%

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

Alternative 15: 55.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		tmp = Float64(Float64(1.0 / 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 (alpha <= 1.0)
		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, 1.0], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 50.5

         \[\leadsto \frac{\frac{1}{\beta}}{\beta} \]

   9. Applied rewrites 50.5%

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

   if 1 < alpha

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      5. lower-/.f64 35.1

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

   9. Applied rewrites 35.1%

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

Alternative 16: 55.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		tmp = Float64(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 (alpha <= 1.0)
		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, 1.0], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 50.1%

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

   if 1 < alpha

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

      5. lower-/.f64 35.1

         \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]

   9. Applied rewrites 35.1%

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

Alternative 17: 52.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], 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

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 50.1%

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

   if 1 < alpha

   1. Initial program 94.4%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

      3. lift-*.f64 32.3

         \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   7. Applied rewrites 32.3%

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

Alternative 18: 32.3% accurate, 6.9× 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta)
use fmin_fmax_functions
    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 94.4%

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

2. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 53.0

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

4. Applied rewrites 53.0%

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

5. Taylor expanded in alpha around inf

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

   3. lift-*.f64 32.3

      \[\leadsto \frac{\alpha}{\beta \cdot \beta} \]

7. Applied rewrites 32.3%

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

Reproduce

herbie shell --seed 2025135 
(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)))