Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.5%

Time: 4.5s

Alternatives: 20

Speedup: 0.1×

Specification

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

Math

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

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: 94.6% accurate, 1.0× speedup

Math

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

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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\\ t_1 := {t\_0}^{2}\\ t_2 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{min}\left(\alpha, \beta\right) \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(t\_2 - -2\right)}^{-2} \cdot 1}{\frac{-1}{\left(\mathsf{min}\left(\alpha, \beta\right) - -1\right) \cdot \mathsf{max}\left(\alpha, \beta\right) - \left(-1 - \mathsf{min}\left(\alpha, \beta\right)\right)} \cdot \left(-3 - \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-1 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \left(-1 \cdot \frac{-2 \cdot \frac{2 + \mathsf{min}\left(\alpha, \beta\right)}{t\_0} - \left(\frac{1}{t\_1} + \frac{\mathsf{min}\left(\alpha, \beta\right)}{t\_1}\right)}{\mathsf{max}\left(\alpha, \beta\right)} + \frac{1}{t\_0}\right)\right)}}{\left(t\_2 + 2 \cdot 1\right) + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (- (* -1.0 (fmin alpha beta)) 1.0))
       (t_1 (pow t_0 2.0))
       (t_2 (+ (fmin alpha beta) (fmax alpha beta))))
  (if (<= (fmin alpha beta) 3.8e-38)
    (/
     (* (pow (- t_2 -2.0) -2.0) 1.0)
     (*
      (/
       -1.0
       (-
        (* (- (fmin alpha beta) -1.0) (fmax alpha beta))
        (- -1.0 (fmin alpha beta))))
      (- -3.0 (+ (fmax alpha beta) (fmin alpha beta)))))
    (/
     (/
      1.0
      (*
       -1.0
       (*
        (fmax alpha beta)
        (+
         (*
          -1.0
          (/
           (-
            (* -2.0 (/ (+ 2.0 (fmin alpha beta)) t_0))
            (+ (/ 1.0 t_1) (/ (fmin alpha beta) t_1)))
           (fmax alpha beta)))
         (/ 1.0 t_0)))))
     (+ (+ t_2 (* 2.0 1.0)) 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-1.0 * fmin(alpha, beta)) - 1.0;
	double t_1 = pow(t_0, 2.0);
	double t_2 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmin(alpha, beta) <= 3.8e-38) {
		tmp = (pow((t_2 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))));
	} else {
		tmp = (1.0 / (-1.0 * (fmax(alpha, beta) * ((-1.0 * (((-2.0 * ((2.0 + fmin(alpha, beta)) / t_0)) - ((1.0 / t_1) + (fmin(alpha, beta) / t_1))) / fmax(alpha, beta))) + (1.0 / t_0))))) / ((t_2 + (2.0 * 1.0)) + 1.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-1.0d0) * fmin(alpha, beta)) - 1.0d0
    t_1 = t_0 ** 2.0d0
    t_2 = fmin(alpha, beta) + fmax(alpha, beta)
    if (fmin(alpha, beta) <= 3.8d-38) then
        tmp = (((t_2 - (-2.0d0)) ** (-2.0d0)) * 1.0d0) / (((-1.0d0) / (((fmin(alpha, beta) - (-1.0d0)) * fmax(alpha, beta)) - ((-1.0d0) - fmin(alpha, beta)))) * ((-3.0d0) - (fmax(alpha, beta) + fmin(alpha, beta))))
    else
        tmp = (1.0d0 / ((-1.0d0) * (fmax(alpha, beta) * (((-1.0d0) * ((((-2.0d0) * ((2.0d0 + fmin(alpha, beta)) / t_0)) - ((1.0d0 / t_1) + (fmin(alpha, beta) / t_1))) / fmax(alpha, beta))) + (1.0d0 / t_0))))) / ((t_2 + (2.0d0 * 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-1.0 * fmin(alpha, beta)) - 1.0;
	double t_1 = Math.pow(t_0, 2.0);
	double t_2 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmin(alpha, beta) <= 3.8e-38) {
		tmp = (Math.pow((t_2 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))));
	} else {
		tmp = (1.0 / (-1.0 * (fmax(alpha, beta) * ((-1.0 * (((-2.0 * ((2.0 + fmin(alpha, beta)) / t_0)) - ((1.0 / t_1) + (fmin(alpha, beta) / t_1))) / fmax(alpha, beta))) + (1.0 / t_0))))) / ((t_2 + (2.0 * 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-1.0 * fmin(alpha, beta)) - 1.0
	t_1 = math.pow(t_0, 2.0)
	t_2 = fmin(alpha, beta) + fmax(alpha, beta)
	tmp = 0
	if fmin(alpha, beta) <= 3.8e-38:
		tmp = (math.pow((t_2 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))))
	else:
		tmp = (1.0 / (-1.0 * (fmax(alpha, beta) * ((-1.0 * (((-2.0 * ((2.0 + fmin(alpha, beta)) / t_0)) - ((1.0 / t_1) + (fmin(alpha, beta) / t_1))) / fmax(alpha, beta))) + (1.0 / t_0))))) / ((t_2 + (2.0 * 1.0)) + 1.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmin(alpha, beta) <= 3.8e-38)
		tmp = Float64(Float64((Float64(t_2 - -2.0) ^ -2.0) * 1.0) / Float64(Float64(-1.0 / Float64(Float64(Float64(fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - Float64(-1.0 - fmin(alpha, beta)))) * Float64(-3.0 - Float64(fmax(alpha, beta) + fmin(alpha, beta)))));
	else
		tmp = Float64(Float64(1.0 / Float64(-1.0 * Float64(fmax(alpha, beta) * Float64(Float64(-1.0 * Float64(Float64(Float64(-2.0 * Float64(Float64(2.0 + fmin(alpha, beta)) / t_0)) - Float64(Float64(1.0 / t_1) + Float64(fmin(alpha, beta) / t_1))) / fmax(alpha, beta))) + Float64(1.0 / t_0))))) / Float64(Float64(t_2 + Float64(2.0 * 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-1.0 * min(alpha, beta)) - 1.0;
	t_1 = t_0 ^ 2.0;
	t_2 = min(alpha, beta) + max(alpha, beta);
	tmp = 0.0;
	if (min(alpha, beta) <= 3.8e-38)
		tmp = (((t_2 - -2.0) ^ -2.0) * 1.0) / ((-1.0 / (((min(alpha, beta) - -1.0) * max(alpha, beta)) - (-1.0 - min(alpha, beta)))) * (-3.0 - (max(alpha, beta) + min(alpha, beta))));
	else
		tmp = (1.0 / (-1.0 * (max(alpha, beta) * ((-1.0 * (((-2.0 * ((2.0 + min(alpha, beta)) / t_0)) - ((1.0 / t_1) + (min(alpha, beta) / t_1))) / max(alpha, beta))) + (1.0 / t_0))))) / ((t_2 + (2.0 * 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[alpha, beta], $MachinePrecision], 3.8e-38], N[(N[(N[Power[N[(t$95$2 - -2.0), $MachinePrecision], -2.0], $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(-1.0 / N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-3.0 - N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(-1.0 * N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(-1.0 * N[(N[(N[(-2.0 * N[(N[(2.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / t$95$1), $MachinePrecision] + N[(N[Min[alpha, beta], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.8e-38

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 94.6%

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 94.6%

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

   3. Applied rewrites 94.6%

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

   5. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. pow-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 93.8%

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 93.8%

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

      18. lower-+.f64 N/A

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

      19. add-flip N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 93.8%

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

   9. Applied rewrites 93.8%

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

   if 3.8e-38 < alpha

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 94.6%

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 94.6%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   6. Applied rewrites 39.6%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{min}\left(\alpha, \beta\right) \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{{\left(t\_0 - -2\right)}^{-2} \cdot 1}{\frac{-1}{\left(\mathsf{min}\left(\alpha, \beta\right) - -1\right) \cdot \mathsf{max}\left(\alpha, \beta\right) - \left(-1 - \mathsf{min}\left(\alpha, \beta\right)\right)} \cdot \left(-3 - \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\right)}{t\_0 + 2 \cdot 1}}{\left(1 + \frac{\mathsf{min}\left(\alpha, \beta\right) - -3}{\mathsf{max}\left(\alpha, \beta\right)}\right) \cdot \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
  (if (<= (fmin alpha beta) 2.35e-32)
    (/
     (* (pow (- t_0 -2.0) -2.0) 1.0)
     (*
      (/
       -1.0
       (-
        (* (- (fmin alpha beta) -1.0) (fmax alpha beta))
        (- -1.0 (fmin alpha beta))))
      (- -3.0 (+ (fmax alpha beta) (fmin alpha beta)))))
    (/
     (/
      (* -1.0 (- (* -1.0 (fmin alpha beta)) 1.0))
      (+ t_0 (* 2.0 1.0)))
     (*
      (+ 1.0 (/ (- (fmin alpha beta) -3.0) (fmax alpha beta)))
      (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmin(alpha, beta) <= 2.35e-32) {
		tmp = (pow((t_0 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / (t_0 + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    if (fmin(alpha, beta) <= 2.35d-32) then
        tmp = (((t_0 - (-2.0d0)) ** (-2.0d0)) * 1.0d0) / (((-1.0d0) / (((fmin(alpha, beta) - (-1.0d0)) * fmax(alpha, beta)) - ((-1.0d0) - fmin(alpha, beta)))) * ((-3.0d0) - (fmax(alpha, beta) + fmin(alpha, beta))))
    else
        tmp = (((-1.0d0) * (((-1.0d0) * fmin(alpha, beta)) - 1.0d0)) / (t_0 + (2.0d0 * 1.0d0))) / ((1.0d0 + ((fmin(alpha, beta) - (-3.0d0)) / fmax(alpha, beta))) * fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmin(alpha, beta) <= 2.35e-32) {
		tmp = (Math.pow((t_0 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / (t_0 + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	tmp = 0
	if fmin(alpha, beta) <= 2.35e-32:
		tmp = (math.pow((t_0 - -2.0), -2.0) * 1.0) / ((-1.0 / (((fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - (-1.0 - fmin(alpha, beta)))) * (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta))))
	else:
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / (t_0 + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmin(alpha, beta) <= 2.35e-32)
		tmp = Float64(Float64((Float64(t_0 - -2.0) ^ -2.0) * 1.0) / Float64(Float64(-1.0 / Float64(Float64(Float64(fmin(alpha, beta) - -1.0) * fmax(alpha, beta)) - Float64(-1.0 - fmin(alpha, beta)))) * Float64(-3.0 - Float64(fmax(alpha, beta) + fmin(alpha, beta)))));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)) / Float64(t_0 + Float64(2.0 * 1.0))) / Float64(Float64(1.0 + Float64(Float64(fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	tmp = 0.0;
	if (min(alpha, beta) <= 2.35e-32)
		tmp = (((t_0 - -2.0) ^ -2.0) * 1.0) / ((-1.0 / (((min(alpha, beta) - -1.0) * max(alpha, beta)) - (-1.0 - min(alpha, beta)))) * (-3.0 - (max(alpha, beta) + min(alpha, beta))));
	else
		tmp = ((-1.0 * ((-1.0 * min(alpha, beta)) - 1.0)) / (t_0 + (2.0 * 1.0))) / ((1.0 + ((min(alpha, beta) - -3.0) / max(alpha, beta))) * max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[alpha, beta], $MachinePrecision], 2.35e-32], N[(N[(N[Power[N[(t$95$0 - -2.0), $MachinePrecision], -2.0], $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[(-1.0 / N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-3.0 - N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(N[(N[Min[alpha, beta], $MachinePrecision] - -3.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.3500000000000001e-32

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 94.6%

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 94.6%

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

   3. Applied rewrites 94.6%

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

   5. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. pow-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 93.8%

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 93.8%

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

      18. lower-+.f64 N/A

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

      19. add-flip N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 93.8%

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

   9. Applied rewrites 93.8%

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

   if 2.3500000000000001e-32 < alpha

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. sum-to-mult N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-/.f64 N/A

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

      16. lower--.f64 46.1%

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

   6. Applied rewrites 46.1%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right) - t\_0}{t\_0 - -2} \cdot \frac{1}{\left(-2 - t\_0\right) \cdot \left(t\_0 - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\right)}{\left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1}}{\left(1 + \frac{\mathsf{min}\left(\alpha, \beta\right) - -3}{\mathsf{max}\left(\alpha, \beta\right)}\right) \cdot \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 4e+141)
    (*
     (/
      (- (- -1.0 (* (fmax alpha beta) (fmin alpha beta))) t_0)
      (- t_0 -2.0))
     (/ 1.0 (* (- -2.0 t_0) (- t_0 -3.0))))
    (/
     (/
      (* -1.0 (- (* -1.0 (fmin alpha beta)) 1.0))
      (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0)))
     (*
      (+ 1.0 (/ (- (fmin alpha beta) -3.0) (fmax alpha beta)))
      (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_0) / (t_0 - -2.0)) * (1.0 / ((-2.0 - t_0) * (t_0 - -3.0)));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / ((fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 4d+141) then
        tmp = ((((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta))) - t_0) / (t_0 - (-2.0d0))) * (1.0d0 / (((-2.0d0) - t_0) * (t_0 - (-3.0d0))))
    else
        tmp = (((-1.0d0) * (((-1.0d0) * fmin(alpha, beta)) - 1.0d0)) / ((fmin(alpha, beta) + fmax(alpha, beta)) + (2.0d0 * 1.0d0))) / ((1.0d0 + ((fmin(alpha, beta) - (-3.0d0)) / fmax(alpha, beta))) * fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_0) / (t_0 - -2.0)) * (1.0 / ((-2.0 - t_0) * (t_0 - -3.0)));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / ((fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 4e+141:
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_0) / (t_0 - -2.0)) * (1.0 / ((-2.0 - t_0) * (t_0 - -3.0)))
	else:
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / ((fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0))) / ((1.0 + ((fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(Float64(Float64(Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta))) - t_0) / Float64(t_0 - -2.0)) * Float64(1.0 / Float64(Float64(-2.0 - t_0) * Float64(t_0 - -3.0))));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)) / Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))) / Float64(Float64(1.0 + Float64(Float64(fmin(alpha, beta) - -3.0) / fmax(alpha, beta))) * fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 4e+141)
		tmp = (((-1.0 - (max(alpha, beta) * min(alpha, beta))) - t_0) / (t_0 - -2.0)) * (1.0 / ((-2.0 - t_0) * (t_0 - -3.0)));
	else
		tmp = ((-1.0 * ((-1.0 * min(alpha, beta)) - 1.0)) / ((min(alpha, beta) + max(alpha, beta)) + (2.0 * 1.0))) / ((1.0 + ((min(alpha, beta) - -3.0) / max(alpha, beta))) * max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(N[(N[(N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(-2.0 - t$95$0), $MachinePrecision] * N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(N[(N[Min[alpha, beta], $MachinePrecision] - -3.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000001e141

   1. Initial program 94.6%

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

   3. Applied rewrites 93.3%

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

   if 4.0000000000000001e141 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. sum-to-mult N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-/.f64 N/A

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

      16. lower--.f64 46.1%

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

   6. Applied rewrites 46.1%

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

Alternative 4: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right) - t\_1}{t\_1 - -2} \cdot \frac{1}{\left(-2 - t\_1\right) \cdot \left(t\_1 - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 4e+141)
    (*
     (/
      (- (- -1.0 (* (fmax alpha beta) (fmin alpha beta))) t_1)
      (- t_1 -2.0))
     (/ 1.0 (* (- -2.0 t_1) (- t_1 -3.0))))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) * (1.0 / ((-2.0 - t_1) * (t_1 - -3.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 4d+141) then
        tmp = ((((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - (-2.0d0))) * (1.0d0 / (((-2.0d0) - t_1) * (t_1 - (-3.0d0))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) * (1.0 / ((-2.0 - t_1) * (t_1 - -3.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 4e+141:
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) * (1.0 / ((-2.0 - t_1) * (t_1 - -3.0)))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(Float64(Float64(Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / Float64(t_1 - -2.0)) * Float64(1.0 / Float64(Float64(-2.0 - t_1) * Float64(t_1 - -3.0))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 4e+141)
		tmp = (((-1.0 - (max(alpha, beta) * min(alpha, beta))) - t_1) / (t_1 - -2.0)) * (1.0 / ((-2.0 - t_1) * (t_1 - -3.0)));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(N[(N[(N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(t$95$1 - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000001e141

   1. Initial program 94.6%

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

   3. Applied rewrites 93.3%

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

   if 4.0000000000000001e141 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 5: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{\left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right) - t\_1}{t\_1 - -2}}{\left(-2 - t\_1\right) \cdot \left(t\_1 - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 3e+110)
    (/
     (/
      (- (- -1.0 (* (fmax alpha beta) (fmin alpha beta))) t_1)
      (- t_1 -2.0))
     (* (- -2.0 t_1) (- t_1 -3.0)))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) / ((-2.0 - t_1) * (t_1 - -3.0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 3d+110) then
        tmp = ((((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - (-2.0d0))) / (((-2.0d0) - t_1) * (t_1 - (-3.0d0)))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) / ((-2.0 - t_1) * (t_1 - -3.0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 3e+110:
		tmp = (((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (t_1 - -2.0)) / ((-2.0 - t_1) * (t_1 - -3.0))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 3e+110)
		tmp = Float64(Float64(Float64(Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / Float64(t_1 - -2.0)) / Float64(Float64(-2.0 - t_1) * Float64(t_1 - -3.0)));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 3e+110)
		tmp = (((-1.0 - (max(alpha, beta) * min(alpha, beta))) - t_1) / (t_1 - -2.0)) / ((-2.0 - t_1) * (t_1 - -3.0));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 3e+110], N[(N[(N[(N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(t$95$1 - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.0000000000000001e110

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 93.3%

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

   if 3.0000000000000001e110 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 6: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right) - t\_1}{\left(-2 - t\_1\right) \cdot \left(\left(t\_1 - -3\right) \cdot \left(\left(t\_0 - -1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 5e+15)
    (/
     (- (- -1.0 (* (fmax alpha beta) (fmin alpha beta))) t_1)
     (* (- -2.0 t_1) (* (- t_1 -3.0) (+ (- t_0 -1.0) 1.0))))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * ((t_0 - -1.0) + 1.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 5d+15) then
        tmp = (((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (((-2.0d0) - t_1) * ((t_1 - (-3.0d0)) * ((t_0 - (-1.0d0)) + 1.0d0)))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * ((t_0 - -1.0) + 1.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 5e+15:
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * ((t_0 - -1.0) + 1.0)))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+15)
		tmp = Float64(Float64(Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / Float64(Float64(-2.0 - t_1) * Float64(Float64(t_1 - -3.0) * Float64(Float64(t_0 - -1.0) + 1.0))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+15)
		tmp = ((-1.0 - (max(alpha, beta) * min(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * ((t_0 - -1.0) + 1.0)));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+15], N[(N[(N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(N[(t$95$1 - -3.0), $MachinePrecision] * N[(N[(t$95$0 - -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. add-flip-rev N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 85.2%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 85.2%

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

   5. Applied rewrites 85.2%

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

   if 5e15 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 7: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right) - t\_1}{\left(-2 - t\_1\right) \cdot \left(\left(t\_1 - -3\right) \cdot \left(t\_1 - -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 5e+15)
    (/
     (- (- -1.0 (* (fmax alpha beta) (fmin alpha beta))) t_1)
     (* (- -2.0 t_1) (* (- t_1 -3.0) (- t_1 -2.0))))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 5d+15) then
        tmp = (((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / (((-2.0d0) - t_1) * ((t_1 - (-3.0d0)) * (t_1 - (-2.0d0))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 5e+15:
		tmp = ((-1.0 - (fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+15)
		tmp = Float64(Float64(Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta))) - t_1) / Float64(Float64(-2.0 - t_1) * Float64(Float64(t_1 - -3.0) * Float64(t_1 - -2.0))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+15)
		tmp = ((-1.0 - (max(alpha, beta) * min(alpha, beta))) - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+15], N[(N[(N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(N[(t$95$1 - -3.0), $MachinePrecision] * N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

   if 5e15 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 8: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_2 := -2 - t\_1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_1 - \left(-1 - \mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right)\right)}{\left(t\_2 \cdot t\_2\right) \cdot \left(t\_1 - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta)))
       (t_2 (- -2.0 t_1)))
  (if (<= (fmax alpha beta) 5e+15)
    (/
     (- t_1 (- -1.0 (* (fmax alpha beta) (fmin alpha beta))))
     (* (* t_2 t_2) (- t_1 -3.0)))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_2 = -2.0 - t_1;
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = (t_1 - (-1.0 - (fmax(alpha, beta) * fmin(alpha, beta)))) / ((t_2 * t_2) * (t_1 - -3.0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    t_2 = (-2.0d0) - t_1
    if (fmax(alpha, beta) <= 5d+15) then
        tmp = (t_1 - ((-1.0d0) - (fmax(alpha, beta) * fmin(alpha, beta)))) / ((t_2 * t_2) * (t_1 - (-3.0d0)))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_2 = -2.0 - t_1;
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = (t_1 - (-1.0 - (fmax(alpha, beta) * fmin(alpha, beta)))) / ((t_2 * t_2) * (t_1 - -3.0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	t_2 = -2.0 - t_1
	tmp = 0
	if fmax(alpha, beta) <= 5e+15:
		tmp = (t_1 - (-1.0 - (fmax(alpha, beta) * fmin(alpha, beta)))) / ((t_2 * t_2) * (t_1 - -3.0))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_2 = Float64(-2.0 - t_1)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+15)
		tmp = Float64(Float64(t_1 - Float64(-1.0 - Float64(fmax(alpha, beta) * fmin(alpha, beta)))) / Float64(Float64(t_2 * t_2) * Float64(t_1 - -3.0)));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	t_2 = -2.0 - t_1;
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+15)
		tmp = (t_1 - (-1.0 - (max(alpha, beta) * min(alpha, beta)))) / ((t_2 * t_2) * (t_1 - -3.0));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 - t$95$1), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+15], N[(N[(t$95$1 - N[(-1.0 - N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(t$95$1 - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

   if 5e15 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 9: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_2 := t\_1 - -2\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t\_1 - -1}{t\_2}}{\left(t\_1 - -3\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta)))
       (t_2 (- t_1 -2.0)))
  (if (<= (fmax alpha beta) 5e+15)
    (/ (/ (- t_1 -1.0) t_2) (* (- t_1 -3.0) t_2))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_2 = t_1 - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((t_1 - -1.0) / t_2) / ((t_1 - -3.0) * t_2);
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    t_2 = t_1 - (-2.0d0)
    if (fmax(alpha, beta) <= 5d+15) then
        tmp = ((t_1 - (-1.0d0)) / t_2) / ((t_1 - (-3.0d0)) * t_2)
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_2 = t_1 - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = ((t_1 - -1.0) / t_2) / ((t_1 - -3.0) * t_2);
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	t_2 = t_1 - -2.0
	tmp = 0
	if fmax(alpha, beta) <= 5e+15:
		tmp = ((t_1 - -1.0) / t_2) / ((t_1 - -3.0) * t_2)
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_2 = Float64(t_1 - -2.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+15)
		tmp = Float64(Float64(Float64(t_1 - -1.0) / t_2) / Float64(Float64(t_1 - -3.0) * t_2));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	t_2 = t_1 - -2.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+15)
		tmp = ((t_1 - -1.0) / t_2) / ((t_1 - -3.0) * t_2);
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -2.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+15], N[(N[(N[(t$95$1 - -1.0), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[(t$95$1 - -3.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 93.8%

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

   if 5e15 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 10: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1 - t\_1}{\left(-2 - t\_1\right) \cdot \left(\left(t\_1 - -3\right) \cdot \left(t\_1 - -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 5e+15)
    (/ (- -1.0 t_1) (* (- -2.0 t_1) (* (- t_1 -3.0) (- t_1 -2.0))))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 5d+15) then
        tmp = ((-1.0d0) - t_1) / (((-2.0d0) - t_1) * ((t_1 - (-3.0d0)) * (t_1 - (-2.0d0))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 5e+15) {
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 5e+15:
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+15)
		tmp = Float64(Float64(-1.0 - t_1) / Float64(Float64(-2.0 - t_1) * Float64(Float64(t_1 - -3.0) * Float64(t_1 - -2.0))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+15)
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((t_1 - -3.0) * (t_1 - -2.0)));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+15], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(N[(t$95$1 - -3.0), $MachinePrecision] * N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e15

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 87.1%

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

   if 5e15 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 11: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 11000000:\\ \;\;\;\;\frac{-1 - t\_1}{\left(-2 - t\_1\right) \cdot \left(\left(2 + \mathsf{min}\left(\alpha, \beta\right)\right) \cdot \left(3 + \mathsf{min}\left(\alpha, \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (+ (fmax alpha beta) (fmin alpha beta))))
  (if (<= (fmax alpha beta) 11000000.0)
    (/
     (- -1.0 t_1)
     (*
      (- -2.0 t_1)
      (* (+ 2.0 (fmin alpha beta)) (+ 3.0 (fmin alpha beta)))))
    (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 11000000.0) {
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((2.0 + fmin(alpha, beta)) * (3.0 + fmin(alpha, beta))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmax(alpha, beta) + fmin(alpha, beta)
    if (fmax(alpha, beta) <= 11000000.0d0) then
        tmp = ((-1.0d0) - t_1) / (((-2.0d0) - t_1) * ((2.0d0 + fmin(alpha, beta)) * (3.0d0 + fmin(alpha, beta))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmax(alpha, beta) + fmin(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 11000000.0) {
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((2.0 + fmin(alpha, beta)) * (3.0 + fmin(alpha, beta))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmax(alpha, beta) + fmin(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 11000000.0:
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((2.0 + fmin(alpha, beta)) * (3.0 + fmin(alpha, beta))))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 11000000.0)
		tmp = Float64(Float64(-1.0 - t_1) / Float64(Float64(-2.0 - t_1) * Float64(Float64(2.0 + fmin(alpha, beta)) * Float64(3.0 + fmin(alpha, beta)))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = max(alpha, beta) + min(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 11000000.0)
		tmp = (-1.0 - t_1) / ((-2.0 - t_1) * ((2.0 + min(alpha, beta)) * (3.0 + min(alpha, beta))));
	else
		tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 11000000.0], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[(N[(-2.0 - t$95$1), $MachinePrecision] * N[(N[(2.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * N[(3.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1e7

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 87.1%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 67.0%

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

   9. Applied rewrites 67.0%

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

   if 1.1e7 < beta

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

      1. metadata-eval 38.4%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 38.4%

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

Alternative 12: 62.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) - -1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{\left(-3 - t\_0\right) \cdot \left(t\_0 - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\mathsf{max}\left(\alpha, \beta\right)}}{t\_0 - -3}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_1 (- (fmin alpha beta) -1.0)))
  (if (<= (fmax alpha beta) 4e+141)
    (* t_1 (/ -1.0 (* (- -3.0 t_0) (- t_0 -2.0))))
    (/ (/ t_1 (fmax alpha beta)) (- t_0 -3.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = t_1 * (-1.0 / ((-3.0 - t_0) * (t_0 - -2.0)));
	} else {
		tmp = (t_1 / fmax(alpha, beta)) / (t_0 - -3.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    t_1 = fmin(alpha, beta) - (-1.0d0)
    if (fmax(alpha, beta) <= 4d+141) then
        tmp = t_1 * ((-1.0d0) / (((-3.0d0) - t_0) * (t_0 - (-2.0d0))))
    else
        tmp = (t_1 / fmax(alpha, beta)) / (t_0 - (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = t_1 * (-1.0 / ((-3.0 - t_0) * (t_0 - -2.0)));
	} else {
		tmp = (t_1 / fmax(alpha, beta)) / (t_0 - -3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	t_1 = fmin(alpha, beta) - -1.0
	tmp = 0
	if fmax(alpha, beta) <= 4e+141:
		tmp = t_1 * (-1.0 / ((-3.0 - t_0) * (t_0 - -2.0)))
	else:
		tmp = (t_1 / fmax(alpha, beta)) / (t_0 - -3.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) - -1.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(t_1 * Float64(-1.0 / Float64(Float64(-3.0 - t_0) * Float64(t_0 - -2.0))));
	else
		tmp = Float64(Float64(t_1 / fmax(alpha, beta)) / Float64(t_0 - -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	t_1 = min(alpha, beta) - -1.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 4e+141)
		tmp = t_1 * (-1.0 / ((-3.0 - t_0) * (t_0 - -2.0)));
	else
		tmp = (t_1 / max(alpha, beta)) / (t_0 - -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(t$95$1 * N[(-1.0 / N[(N[(-3.0 - t$95$0), $MachinePrecision] * N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000001e141

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

   6. Applied rewrites 49.3%

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

   if 4.0000000000000001e141 < beta

   1. Initial program 94.6%

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

   2. 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 30.4%

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

   4. Applied rewrites 30.4%

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

      1. metadata-eval 30.4%

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

      2. metadata-eval 30.4%

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 30.4%

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

      1. metadata-eval 30.4%

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

      2. metadata-eval 30.4%

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 30.4%

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

Alternative 13: 62.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) - -1\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := t\_1 - -3\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{t\_0}{t\_2 \cdot \left(t\_1 - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{max}\left(\alpha, \beta\right)}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (- (fmin alpha beta) -1.0))
       (t_1 (+ (fmin alpha beta) (fmax alpha beta)))
       (t_2 (- t_1 -3.0)))
  (if (<= (fmax alpha beta) 3e+110)
    (/ t_0 (* t_2 (- t_1 -2.0)))
    (/ (/ t_0 (fmax alpha beta)) t_2))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) - -1.0;
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 - -3.0;
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = t_0 / (t_2 * (t_1 - -2.0));
	} else {
		tmp = (t_0 / fmax(alpha, beta)) / t_2;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(alpha, beta) - (-1.0d0)
    t_1 = fmin(alpha, beta) + fmax(alpha, beta)
    t_2 = t_1 - (-3.0d0)
    if (fmax(alpha, beta) <= 3d+110) then
        tmp = t_0 / (t_2 * (t_1 - (-2.0d0)))
    else
        tmp = (t_0 / fmax(alpha, beta)) / t_2
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) - -1.0;
	double t_1 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_2 = t_1 - -3.0;
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = t_0 / (t_2 * (t_1 - -2.0));
	} else {
		tmp = (t_0 / fmax(alpha, beta)) / t_2;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) - -1.0
	t_1 = fmin(alpha, beta) + fmax(alpha, beta)
	t_2 = t_1 - -3.0
	tmp = 0
	if fmax(alpha, beta) <= 3e+110:
		tmp = t_0 / (t_2 * (t_1 - -2.0))
	else:
		tmp = (t_0 / fmax(alpha, beta)) / t_2
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) - -1.0)
	t_1 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_2 = Float64(t_1 - -3.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 3e+110)
		tmp = Float64(t_0 / Float64(t_2 * Float64(t_1 - -2.0)));
	else
		tmp = Float64(Float64(t_0 / fmax(alpha, beta)) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) - -1.0;
	t_1 = min(alpha, beta) + max(alpha, beta);
	t_2 = t_1 - -3.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 3e+110)
		tmp = t_0 / (t_2 * (t_1 - -2.0));
	else
		tmp = (t_0 / max(alpha, beta)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -3.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 3e+110], N[(t$95$0 / N[(t$95$2 * N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.0000000000000001e110

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

   6. Applied rewrites 49.3%

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

   if 3.0000000000000001e110 < beta

   1. Initial program 94.6%

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

   2. 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 30.4%

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

   4. Applied rewrites 30.4%

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

      1. metadata-eval 30.4%

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

      2. metadata-eval 30.4%

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 30.4%

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

      1. metadata-eval 30.4%

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

      2. metadata-eval 30.4%

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 30.4%

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

Alternative 14: 62.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 3 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\left(t\_0 - -3\right) \cdot \left(t\_0 - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{min}\left(\alpha, \beta\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
  (if (<= (fmax alpha beta) 3e+110)
    (/ (- (fmin alpha beta) -1.0) (* (- t_0 -3.0) (- t_0 -2.0)))
    (/
     (/ (+ 1.0 (fmin alpha beta)) (fmax alpha beta))
     (+ 3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = (fmin(alpha, beta) - -1.0) / ((t_0 - -3.0) * (t_0 - -2.0));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    if (fmax(alpha, beta) <= 3d+110) then
        tmp = (fmin(alpha, beta) - (-1.0d0)) / ((t_0 - (-3.0d0)) * (t_0 - (-2.0d0)))
    else
        tmp = ((1.0d0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0d0 + fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 3e+110) {
		tmp = (fmin(alpha, beta) - -1.0) / ((t_0 - -3.0) * (t_0 - -2.0));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 3e+110:
		tmp = (fmin(alpha, beta) - -1.0) / ((t_0 - -3.0) * (t_0 - -2.0))
	else:
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 3e+110)
		tmp = Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(Float64(t_0 - -3.0) * Float64(t_0 - -2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / Float64(3.0 + fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 3e+110)
		tmp = (min(alpha, beta) - -1.0) / ((t_0 - -3.0) * (t_0 - -2.0));
	else
		tmp = ((1.0 + min(alpha, beta)) / max(alpha, beta)) / (3.0 + max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 3e+110], N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(t$95$0 - -3.0), $MachinePrecision] * N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.0000000000000001e110

   1. Initial program 94.6%

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

   2. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 38.4%

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

   4. Applied rewrites 38.4%

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

   6. Applied rewrites 49.3%

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

   if 3.0000000000000001e110 < beta

   1. Initial program 94.6%

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

   2. 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 30.4%

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 30.3%

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

   7. Applied rewrites 30.3%

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

Alternative 15: 62.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{t\_0 - -3}}{t\_0 - -2} \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
  (/ (/ (- (fmin alpha beta) -1.0) (- t_0 -3.0)) (- t_0 -2.0))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	return ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
}

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 = fmin(alpha, beta) + fmax(alpha, beta)
    code = ((fmin(alpha, beta) - (-1.0d0)) / (t_0 - (-3.0d0))) / (t_0 - (-2.0d0))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	return ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	return ((fmin(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0)

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	return Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / Float64(t_0 - -3.0)) / Float64(t_0 - -2.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	tmp = ((min(alpha, beta) - -1.0) / (t_0 - -3.0)) / (t_0 - -2.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 94.6%

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

2. 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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 38.4%

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

4. Applied rewrites 38.4%

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

   1. metadata-eval 38.4%

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

6. Applied rewrites 38.4%

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

Alternative 16: 62.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 11000000:\\ \;\;\;\;\frac{1 + \mathsf{max}\left(\alpha, \beta\right)}{\left(t\_0 - -3\right) \cdot \left(t\_0 - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{min}\left(\alpha, \beta\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
  (if (<= (fmax alpha beta) 11000000.0)
    (/ (+ 1.0 (fmax alpha beta)) (* (- t_0 -3.0) (- t_0 -2.0)))
    (/
     (/ (+ 1.0 (fmin alpha beta)) (fmax alpha beta))
     (+ 3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 11000000.0) {
		tmp = (1.0 + fmax(alpha, beta)) / ((t_0 - -3.0) * (t_0 - -2.0));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = fmin(alpha, beta) + fmax(alpha, beta)
    if (fmax(alpha, beta) <= 11000000.0d0) then
        tmp = (1.0d0 + fmax(alpha, beta)) / ((t_0 - (-3.0d0)) * (t_0 - (-2.0d0)))
    else
        tmp = ((1.0d0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0d0 + fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 11000000.0) {
		tmp = (1.0 + fmax(alpha, beta)) / ((t_0 - -3.0) * (t_0 - -2.0));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) + fmax(alpha, beta)
	tmp = 0
	if fmax(alpha, beta) <= 11000000.0:
		tmp = (1.0 + fmax(alpha, beta)) / ((t_0 - -3.0) * (t_0 - -2.0))
	else:
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 11000000.0)
		tmp = Float64(Float64(1.0 + fmax(alpha, beta)) / Float64(Float64(t_0 - -3.0) * Float64(t_0 - -2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / Float64(3.0 + fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) + max(alpha, beta);
	tmp = 0.0;
	if (max(alpha, beta) <= 11000000.0)
		tmp = (1.0 + max(alpha, beta)) / ((t_0 - -3.0) * (t_0 - -2.0));
	else
		tmp = ((1.0 + min(alpha, beta)) / max(alpha, beta)) / (3.0 + max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 11000000.0], N[(N[(1.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 - -3.0), $MachinePrecision] * N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1e7

   1. Initial program 94.6%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 36.7%

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

   4. Applied rewrites 36.7%

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

      1. metadata-eval 36.7%

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 48.7%

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

   if 1.1e7 < beta

   1. Initial program 94.6%

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

   2. 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 30.4%

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 30.3%

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

   7. Applied rewrites 30.3%

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

Alternative 17: 56.5% accurate, 0.3× speedup

Math

\[\frac{\frac{1 + \mathsf{min}\left(\alpha, \beta\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{max}\left(\alpha, \beta\right)} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (/
 (/ (+ 1.0 (fmin alpha beta)) (fmax alpha beta))
 (+ 3.0 (fmax alpha beta))))

C

double code(double alpha, double beta) {
	return ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
}

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
    code = ((1.0d0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0d0 + fmax(alpha, beta))
end function

Java

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

Python

def code(alpha, beta):
	return ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta))

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / Float64(3.0 + fmax(alpha, beta)))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = ((1.0 + min(alpha, beta)) / max(alpha, beta)) / (3.0 + max(alpha, beta));
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(1.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

2. 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 30.4%

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

4. Applied rewrites 30.4%

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

5. Taylor expanded in alpha around 0

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

   1. lower-+.f64 30.3%

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

7. Applied rewrites 30.3%

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

Alternative 18: 51.5% accurate, 0.4× speedup

Math

\[\frac{\frac{1}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{max}\left(\alpha, \beta\right)} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (/ (/ 1.0 (fmax alpha beta)) (+ 3.0 (fmax alpha beta))))

C

double code(double alpha, double beta) {
	return (1.0 / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
}

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
    code = (1.0d0 / fmax(alpha, beta)) / (3.0d0 + fmax(alpha, beta))
end function

Java

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

Python

def code(alpha, beta):
	return (1.0 / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta))

Julia

function code(alpha, beta)
	return Float64(Float64(1.0 / fmax(alpha, beta)) / Float64(3.0 + fmax(alpha, beta)))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (1.0 / max(alpha, beta)) / (3.0 + max(alpha, beta));
end

Wolfram

code[alpha_, beta_] := N[(N[(1.0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

2. 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 30.4%

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

4. Applied rewrites 30.4%

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

5. Taylor expanded in beta around 0

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

   1. lower-+.f64 4.3%

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

7. Applied rewrites 4.3%

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

8. Taylor expanded in alpha around 0

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

   1. lower-/.f64 7.7%

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

10. Applied rewrites 7.7%

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

11. Taylor expanded in alpha around 0

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

    1. lower-+.f64 28.5%

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

13. Applied rewrites 28.5%

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

Alternative 19: 9.0% accurate, 0.4× speedup

Math

\[\frac{\frac{1}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{min}\left(\alpha, \beta\right)} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (/ (/ 1.0 (fmax alpha beta)) (+ 3.0 (fmin alpha beta))))

C

double code(double alpha, double beta) {
	return (1.0 / fmax(alpha, beta)) / (3.0 + fmin(alpha, beta));
}

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
    code = (1.0d0 / fmax(alpha, beta)) / (3.0d0 + fmin(alpha, beta))
end function

Java

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

Python

def code(alpha, beta):
	return (1.0 / fmax(alpha, beta)) / (3.0 + fmin(alpha, beta))

Julia

function code(alpha, beta)
	return Float64(Float64(1.0 / fmax(alpha, beta)) / Float64(3.0 + fmin(alpha, beta)))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (1.0 / max(alpha, beta)) / (3.0 + min(alpha, beta));
end

Wolfram

code[alpha_, beta_] := N[(N[(1.0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

2. 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 30.4%

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

4. Applied rewrites 30.4%

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

5. Taylor expanded in beta around 0

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

   1. lower-+.f64 4.3%

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

7. Applied rewrites 4.3%

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

8. Taylor expanded in alpha around 0

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

   1. lower-/.f64 7.7%

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

10. Applied rewrites 7.7%

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

Alternative 20: 6.1% accurate, 0.7× speedup

Math

\[\frac{\frac{1}{\mathsf{max}\left(\alpha, \beta\right)}}{3} \]

FPCore

(FPCore (alpha beta)
  :precision binary64
  (/ (/ 1.0 (fmax alpha beta)) 3.0))

C

double code(double alpha, double beta) {
	return (1.0 / fmax(alpha, beta)) / 3.0;
}

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
    code = (1.0d0 / fmax(alpha, beta)) / 3.0d0
end function

Java

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

Python

def code(alpha, beta):
	return (1.0 / fmax(alpha, beta)) / 3.0

Julia

function code(alpha, beta)
	return Float64(Float64(1.0 / fmax(alpha, beta)) / 3.0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (1.0 / max(alpha, beta)) / 3.0;
end

Wolfram

code[alpha_, beta_] := N[(N[(1.0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

2. 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 30.4%

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

4. Applied rewrites 30.4%

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

5. Taylor expanded in beta around 0

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

   1. lower-+.f64 4.3%

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

7. Applied rewrites 4.3%

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

8. Taylor expanded in alpha around 0

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

   1. lower-/.f64 7.7%

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

10. Applied rewrites 7.7%

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

11. Taylor expanded in alpha around 0

    \[\leadsto \frac{\frac{1}{\beta}}{3} \]
12. Step-by-step derivation

13. Applied rewrites 4.4%

    \[\leadsto \frac{\frac{1}{\beta}}{3} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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)))