Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.7%

Time: 4.8s

Alternatives: 18

Speedup: 3.5×

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: 94.7% 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

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\mathsf{fma}\left(4 + \left(\beta + \left(\alpha + \alpha\right)\right), \beta, \left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (if (<= beta 3.4e+65)
     (/
      (/
       (+ (fma beta alpha (+ beta alpha)) 1.0)
       (fma
        (+ 4.0 (+ beta (+ alpha alpha)))
        beta
        (* (+ 2.0 alpha) (+ 2.0 alpha))))
      (+ 3.0 (+ beta alpha)))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       t_0)
      (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double tmp;
	if (beta <= 3.4e+65) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / fma((4.0 + (beta + (alpha + alpha))), beta, ((2.0 + alpha) * (2.0 + alpha)))) / (3.0 + (beta + alpha));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	tmp = 0.0
	if (beta <= 3.4e+65)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / fma(Float64(4.0 + Float64(beta + Float64(alpha + alpha))), beta, 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(Float64(2.0 + alpha) / beta))) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.4e+65], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(4.0 + N[(beta + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + 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), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3999999999999999e65

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if 3.3999999999999999e65 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. div-add N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. div-add N/A

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

      18. lower-/.f64 N/A

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

   4. Applied rewrites 55.4%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\mathsf{fma}\left(4 + \left(\beta + \left(\alpha + \alpha\right)\right), \beta, \left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\left(\alpha + \frac{1 + \alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{4 + \left(\alpha + \alpha\right)}{\beta}\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= beta 3.4e+65)
     (/
      (/
       (+ (fma beta alpha (+ beta alpha)) 1.0)
       (fma
        (+ 4.0 (+ beta (+ alpha alpha)))
        beta
        (* (+ 2.0 alpha) (+ 2.0 alpha))))
      t_0)
     (/
      (/
       (+
        1.0
        (-
         (+ alpha (/ (+ 1.0 alpha) beta))
         (* (+ 1.0 alpha) (/ (+ 4.0 (+ alpha alpha)) beta))))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 3.4e+65) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / fma((4.0 + (beta + (alpha + alpha))), beta, ((2.0 + alpha) * (2.0 + alpha)))) / t_0;
	} else {
		tmp = ((1.0 + ((alpha + ((1.0 + alpha) / beta)) - ((1.0 + alpha) * ((4.0 + (alpha + alpha)) / beta)))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 3.4e+65)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / fma(Float64(4.0 + Float64(beta + Float64(alpha + alpha))), beta, Float64(Float64(2.0 + alpha) * Float64(2.0 + alpha)))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(alpha + Float64(Float64(1.0 + alpha) / beta)) - Float64(Float64(1.0 + alpha) * Float64(Float64(4.0 + Float64(alpha + alpha)) / beta)))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.4e+65], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(4.0 + N[(beta + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(alpha + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(4.0 + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3999999999999999e65

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if 3.3999999999999999e65 < beta

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(t\_1 \cdot t\_1\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\left(\alpha + \frac{1 + \alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{4 + \left(\alpha + \alpha\right)}{\beta}\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.4e+16)
     (/ (+ (fma alpha beta (+ beta alpha)) 1.0) (* (* t_1 t_1) t_0))
     (/
      (/
       (+
        1.0
        (-
         (+ alpha (/ (+ 1.0 alpha) beta))
         (* (+ 1.0 alpha) (/ (+ 4.0 (+ alpha alpha)) beta))))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.4e+16) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / ((t_1 * t_1) * t_0);
	} else {
		tmp = ((1.0 + ((alpha + ((1.0 + alpha) / beta)) - ((1.0 + alpha) * ((4.0 + (alpha + alpha)) / beta)))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.4e+16)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(Float64(t_1 * t_1) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(alpha + Float64(Float64(1.0 + alpha) / beta)) - Float64(Float64(1.0 + alpha) * Float64(Float64(4.0 + Float64(alpha + alpha)) / beta)))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.4e+16], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(alpha + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(4.0 + N[(alpha + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4e16

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 85.3%

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

   if 1.4e16 < beta

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

Alternative 4: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(t\_0 \cdot t\_0\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.4e+16)
     (/
      (+ (fma alpha beta (+ beta alpha)) 1.0)
      (* (* t_0 t_0) (+ 3.0 (+ beta alpha))))
     (/
      (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
      (+ (+ alpha beta) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.4e+16) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / ((t_0 * t_0) * (3.0 + (beta + alpha)));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	}
	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 <= 1.4e+16)
		tmp = Float64(Float64(fma(alpha, beta, 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(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.4e+16], N[(N[(N[(alpha * beta + 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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4e16

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 85.3%

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

   if 1.4e16 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. metadata-eval 62.2

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

      2. metadata-eval 62.2

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + \mathsf{Rewrite=>}\left(metadata-eval, 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5e+15)
   (/
    (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 2.0 beta)))
    (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5e+15) {
		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)) / ((alpha + beta) + 3.0);
	}
	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 <= 8.5d+15) 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)) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5e+15) {
		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)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5e+15:
		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)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5e+15)
		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(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5e+15)
		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)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5e+15], 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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5e15

   1. Initial program 94.7%

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

   2. 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 93.5

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

   4. Applied rewrites 93.5%

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

   if 8.5e15 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. metadata-eval 62.2

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

      2. metadata-eval 62.2

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + \mathsf{Rewrite=>}\left(metadata-eval, 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

Alternative 6: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5)
   (/ (/ (+ 1.0 alpha) (* (+ 2.0 alpha) (+ 2.0 alpha))) (+ 3.0 (+ beta alpha)))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (2.0 + alpha))) / (3.0 + (beta + alpha));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	}
	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 <= 9.5d0) then
        tmp = ((1.0d0 + alpha) / ((2.0d0 + alpha) * (2.0d0 + alpha))) / (3.0d0 + (beta + alpha))
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (2.0 + alpha))) / (3.0 + (beta + alpha));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.5:
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (2.0 + alpha))) / (3.0 + (beta + alpha))
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(Float64(Float64(1.0 + 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(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5)
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (2.0 + alpha))) / (3.0 + (beta + alpha));
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.5], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   1. Initial program 94.7%

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

   3. Applied rewrites 93.3%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

   if 9.5 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. metadata-eval 62.2

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

      2. metadata-eval 62.2

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + \mathsf{Rewrite=>}\left(metadata-eval, 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

Alternative 7: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2e+15)
   (/ (+ 1.0 beta) (+ 12.0 (* beta (+ 16.0 (* beta (+ 7.0 beta))))))
   (/
    (/ (- (- (- alpha) 1.0)) (+ (+ alpha beta) 2.0))
    (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+15) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	}
	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 <= 2.2d+15) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * (16.0d0 + (beta * (7.0d0 + beta)))))
    else
        tmp = (-(-alpha - 1.0d0) / ((alpha + beta) + 2.0d0)) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+15) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	} else {
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.2e+15:
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))))
	else:
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(7.0 + beta))))));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(-alpha) - 1.0)) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2e+15)
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	else
		tmp = (-(-alpha - 1.0) / ((alpha + beta) + 2.0)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(7.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[((-alpha) - 1.0), $MachinePrecision]) / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2e15

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \color{blue}{\beta \cdot \left(7 + \beta\right)}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(7 + \beta\right)}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \color{blue}{\beta}\right)\right)} \]

      5. lower-+.f64 86.7

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)} \]

   7. Applied rewrites 86.7%

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

   if 2.2e15 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-neg.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. metadata-eval 62.2

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

      2. metadata-eval 62.2

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\left(-\alpha\right) - 1\right)}{\left(\alpha + \beta\right) + \mathsf{Rewrite=>}\left(metadata-eval, 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

Alternative 8: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.66 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.66e+16)
   (/ (+ 1.0 beta) (+ 12.0 (* beta (+ 16.0 (* beta (+ 7.0 beta))))))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66e+16) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	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 <= 1.66d+16) then
        tmp = (1.0d0 + beta) / (12.0d0 + (beta * (16.0d0 + (beta * (7.0d0 + beta)))))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66e+16) {
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.66e+16:
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.66e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(12.0 + Float64(beta * Float64(16.0 + Float64(beta * Float64(7.0 + beta))))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.66e+16)
		tmp = (1.0 + beta) / (12.0 + (beta * (16.0 + (beta * (7.0 + beta)))));
	else
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.66e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(12.0 + N[(beta * N[(16.0 + N[(beta * N[(7.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.66e16

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \color{blue}{\beta \cdot \left(7 + \beta\right)}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(7 + \beta\right)}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \color{blue}{\beta}\right)\right)} \]

      5. lower-+.f64 86.7

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)} \]

   7. Applied rewrites 86.7%

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

   if 1.66e16 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 55.8

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

   6. Applied rewrites 55.8%

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

Alternative 9: 97.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (+
    0.08333333333333333
    (*
     beta
     (-
      (* beta (- (* 0.024691358024691357 beta) 0.011574074074074073))
      0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((0.024691358024691357 * beta) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	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 <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((0.024691358024691357d0 * beta) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((0.024691358024691357 * beta) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.08333333333333333 + (beta * ((beta * ((0.024691358024691357 * beta) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(0.024691358024691357 * beta) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.08333333333333333 + (beta * ((beta * ((0.024691358024691357 * beta) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(0.024691358024691357 * beta), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      6. lower-*.f64 45.0

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right) \]

   7. Applied rewrites 45.0%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 55.8

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

   6. Applied rewrites 55.8%

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

Alternative 10: 97.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (+
    0.08333333333333333
    (* beta (- (* -0.011574074074074073 beta) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	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 <= 1.55d0) then
        tmp = 0.08333333333333333d0 + (beta * (((-0.011574074074074073d0) * beta) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(-0.011574074074074073 * beta) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(0.08333333333333333 + N[(beta * N[(N[(-0.011574074074074073 * beta), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 44.5

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 44.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.55000000000000004 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 55.8

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

   6. Applied rewrites 55.8%

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

Alternative 11: 97.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (+
    0.08333333333333333
    (* beta (- (* -0.011574074074074073 beta) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (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 <= 1.55d0) then
        tmp = 0.08333333333333333d0 + (beta * (((-0.011574074074074073d0) * beta) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (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 <= 1.55) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(-0.011574074074074073 * beta) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / 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 <= 1.55)
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / (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, 1.55], N[(0.08333333333333333 + N[(beta * N[(N[(-0.011574074074074073 * beta), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 44.5

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 44.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.55000000000000004 < beta

   1. Initial program 94.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in alpha around 0

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

      1. lift-+.f64 55.8

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

   7. Applied rewrites 55.8%

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

Alternative 12: 97.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65)
   (+
    0.08333333333333333
    (* beta (- (* -0.011574074074074073 beta) 0.027777777777777776)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + 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 <= 1.65d0) then
        tmp = 0.08333333333333333d0 + (beta * (((-0.011574074074074073d0) * beta) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.65:
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(-0.011574074074074073 * beta) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.65)
		tmp = 0.08333333333333333 + (beta * ((-0.011574074074074073 * beta) - 0.027777777777777776));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.65], N[(0.08333333333333333 + N[(beta * N[(N[(-0.011574074074074073 * beta), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 44.5

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 44.5%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.6499999999999999 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 55.6

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

   6. Applied rewrites 55.6%

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

Alternative 13: 97.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 - 0.027777777777777776 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (- 0.08333333333333333 (* 0.027777777777777776 beta))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else {
		tmp = ((1.0 + 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 <= 2.8d0) then
        tmp = 0.08333333333333333d0 - (0.027777777777777776d0 * beta)
    else
        tmp = ((1.0d0 + 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 <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta)
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 - Float64(0.027777777777777776 * beta));
	else
		tmp = Float64(Float64(Float64(1.0 + 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 <= 2.8)
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	else
		tmp = ((1.0 + 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, 2.8], N[(0.08333333333333333 - N[(0.027777777777777776 * beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \beta} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{12} - \frac{1}{36} \cdot \beta \]

      4. lower-*.f64 44.6

         \[\leadsto 0.08333333333333333 - 0.027777777777777776 \cdot \beta \]

   7. Applied rewrites 44.6%

      \[\leadsto 0.08333333333333333 - \color{blue}{0.027777777777777776 \cdot \beta} \]

   if 2.7999999999999998 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 55.6

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

   6. Applied rewrites 55.6%

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

Alternative 14: 96.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 - 0.027777777777777776 \cdot \beta\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (- 0.08333333333333333 (* 0.027777777777777776 beta))
   (if (<= beta 1.35e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
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 <= 2.8d0) then
        tmp = 0.08333333333333333d0 - (0.027777777777777776d0 * beta)
    else if (beta <= 1.35d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta)
	elif beta <= 1.35e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 - Float64(0.027777777777777776 * beta));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	elseif (beta <= 1.35e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 - N[(0.027777777777777776 * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], 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 < 2.7999999999999998

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \beta} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{12} - \frac{1}{36} \cdot \beta \]

      4. lower-*.f64 44.6

         \[\leadsto 0.08333333333333333 - 0.027777777777777776 \cdot \beta \]

   7. Applied rewrites 44.6%

      \[\leadsto 0.08333333333333333 - \color{blue}{0.027777777777777776 \cdot \beta} \]

   if 2.7999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   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 32.0

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

   7. Applied rewrites 32.0%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   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 34.5

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

   9. Applied rewrites 34.5%

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

Alternative 15: 94.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 - 0.027777777777777776 \cdot \beta\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (- 0.08333333333333333 (* 0.027777777777777776 beta))
   (if (<= beta 1.35e+154) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else if (beta <= 1.35e+154) {
		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 (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 - (0.027777777777777776d0 * beta)
    else if (beta <= 1.35d+154) 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 (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else if (beta <= 1.35e+154) {
		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 beta <= 2.8:
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta)
	elif beta <= 1.35e+154:
		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 (beta <= 2.8)
		tmp = Float64(0.08333333333333333 - Float64(0.027777777777777776 * beta));
	elseif (beta <= 1.35e+154)
		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 (beta <= 2.8)
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	elseif (beta <= 1.35e+154)
		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[beta, 2.8], N[(0.08333333333333333 - N[(0.027777777777777776 * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.7999999999999998

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \beta} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{12} - \frac{1}{36} \cdot \beta \]

      4. lower-*.f64 44.6

         \[\leadsto 0.08333333333333333 - 0.027777777777777776 \cdot \beta \]

   7. Applied rewrites 44.6%

      \[\leadsto 0.08333333333333333 - \color{blue}{0.027777777777777776 \cdot \beta} \]

   if 2.7999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   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 1.35000000000000003e154 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   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 32.0

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

   7. Applied rewrites 32.0%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
   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 34.5

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

   9. Applied rewrites 34.5%

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

Alternative 16: 91.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 - 0.027777777777777776 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (- 0.08333333333333333 (* 0.027777777777777776 beta))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else {
		tmp = 1.0 / (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 <= 2.8d0) then
        tmp = 0.08333333333333333d0 - (0.027777777777777776d0 * beta)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 - Float64(0.027777777777777776 * beta));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 - N[(0.027777777777777776 * beta), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \beta} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{12} - \frac{1}{36} \cdot \beta \]

      4. lower-*.f64 44.6

         \[\leadsto 0.08333333333333333 - 0.027777777777777776 \cdot \beta \]

   7. Applied rewrites 44.6%

      \[\leadsto 0.08333333333333333 - \color{blue}{0.027777777777777776 \cdot \beta} \]

   if 2.7999999999999998 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   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%

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

Alternative 17: 74.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333 - 0.027777777777777776 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (- 0.08333333333333333 (* 0.027777777777777776 beta))
   (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * 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 <= 3.0d0) then
        tmp = 0.08333333333333333d0 - (0.027777777777777776d0 * 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 <= 3.0) {
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.0:
		tmp = 0.08333333333333333 - (0.027777777777777776 * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = Float64(0.08333333333333333 - Float64(0.027777777777777776 * 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 (beta <= 3.0)
		tmp = 0.08333333333333333 - (0.027777777777777776 * 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, 3.0], N[(0.08333333333333333 - N[(0.027777777777777776 * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 94.7%

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

   2. 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 86.7

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \beta} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{12} - \left(\mathsf{neg}\left(\frac{-1}{36}\right)\right) \cdot \color{blue}{\beta} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{12} - \frac{1}{36} \cdot \beta \]

      4. lower-*.f64 44.6

         \[\leadsto 0.08333333333333333 - 0.027777777777777776 \cdot \beta \]

   7. Applied rewrites 44.6%

      \[\leadsto 0.08333333333333333 - \color{blue}{0.027777777777777776 \cdot \beta} \]

   if 3 < beta

   1. Initial program 94.7%

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

   2. 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 52.8

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

   4. Applied rewrites 52.8%

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

   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 32.0

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

   7. Applied rewrites 32.0%

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

Alternative 18: 45.5% accurate, 50.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

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 = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.7%

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

2. 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 86.7

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

4. Applied rewrites 86.7%

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

5. Taylor expanded in beta around 0

   \[\leadsto \frac{1}{12} \]
6. Step-by-step derivation

7. Applied rewrites 45.5%

   \[\leadsto 0.08333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(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)))