Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.6%

Time: 5.6s

Alternatives: 14

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{\left(\frac{2 + \alpha}{\beta} + 1\right) \cdot \beta + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 1e+60)
     (/
      (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)
      (+ (* (+ (/ (+ 2.0 alpha) beta) 1.0) beta) 1.0))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 1e+60) {
		tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (((((2.0 + alpha) / beta) + 1.0) * beta) + 1.0);
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 1e+60)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(Float64(Float64(Float64(Float64(2.0 + alpha) / beta) + 1.0) * beta) + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+60], 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[(N[(N[(N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision] * beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999995e59

   1. Initial program 98.8%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.9999999999999995e59 < beta

   1. Initial program 73.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\frac{2 + \alpha}{\beta} + 1\right) \cdot \beta + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0}}{t\_0}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+60)
     (/
      (/ (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_0) t_0)
      (+ 3.0 (+ beta alpha)))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      (+ (+ (+ alpha beta) 2.0) 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+60) {
		tmp = (((fma(beta, alpha, (beta + alpha)) + 1.0) / t_0) / t_0) / (3.0 + (beta + alpha));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (((alpha + beta) + 2.0) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+60)
		tmp = Float64(Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_0) / t_0) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(Float64(Float64(alpha + beta) + 2.0) + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+60], N[(N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999995e59

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.8

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 98.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.8

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 98.8

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

      18. lift-+.f64 N/A

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

   4. Applied rewrites 98.8%

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

   if 9.9999999999999995e59 < beta

   1. Initial program 73.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_1}}{t\_1}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+60)
     (/
      (/ (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_1) t_1)
      (+ 3.0 (+ beta alpha)))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+60) {
		tmp = (((fma(beta, alpha, (beta + alpha)) + 1.0) / t_1) / t_1) / (3.0 + (beta + alpha));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+60)
		tmp = Float64(Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_1) / t_1) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+60], N[(N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999995e59

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.8

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

      11. lift-*.f64 N/A

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

      12. metadata-eval 98.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.8

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

      16. lift-*.f64 N/A

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

      17. metadata-eval 98.8

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

      18. lift-+.f64 N/A

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

   4. Applied rewrites 98.8%

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

   if 9.9999999999999995e59 < beta

   1. Initial program 73.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. metadata-eval 92.0

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

      2. metadata-eval 92.0

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   7. Applied rewrites 92.0%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+60}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_1}}{t\_1 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+60)
     (/
      (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_1)
      (* t_1 (+ 3.0 (+ beta alpha))))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+60) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / t_1) / (t_1 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+60)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_1) / Float64(t_1 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+60], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999995e59

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 98.6%

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

   if 9.9999999999999995e59 < beta

   1. Initial program 73.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. metadata-eval 92.0

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

      2. metadata-eval 92.0

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   7. Applied rewrites 92.0%

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

Alternative 5: 98.7% accurate, 1.5× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 4e+16)
     (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) t_0) (+ 3.0 beta))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+16) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / (3.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
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 = (alpha + beta) + 2.0d0
    if (beta <= 4d+16) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / t_0) / (3.0d0 + beta)
    else
        tmp = ((1.0d0 + alpha) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+16) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / (3.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 4e+16:
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / (3.0 + beta)
	else:
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 4e+16)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / t_0) / Float64(3.0 + beta));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 4e+16)
		tmp = (((1.0 + beta) / (2.0 + beta)) / t_0) / (3.0 + beta);
	else
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+16], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 86.3

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-+.f64 69.5

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

   11. Applied rewrites 69.5%

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

   if 4e16 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

      1. metadata-eval 87.5

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

      2. metadata-eval 87.5

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   7. Applied rewrites 87.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{2 + \beta}}{\left(\alpha + \beta\right) + 2}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.6× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 4e+16)
     (/ (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (+ 2.0 beta)) (+ 3.0 beta))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+16) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (2.0 + beta)) / (3.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
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 = (alpha + beta) + 2.0d0
    if (beta <= 4d+16) then
        tmp = (((1.0d0 + beta) / (2.0d0 + beta)) / (2.0d0 + beta)) / (3.0d0 + beta)
    else
        tmp = ((1.0d0 + alpha) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+16) {
		tmp = (((1.0 + beta) / (2.0 + beta)) / (2.0 + beta)) / (3.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 4e+16:
		tmp = (((1.0 + beta) / (2.0 + beta)) / (2.0 + beta)) / (3.0 + beta)
	else:
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 4e+16)
		tmp = Float64(Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(2.0 + beta)) / Float64(3.0 + beta));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 4e+16)
		tmp = (((1.0 + beta) / (2.0 + beta)) / (2.0 + beta)) / (3.0 + beta);
	else
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+16], N[(N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 86.3

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in alpha around 0

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

       1. lift-+.f64 81.0

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

   11. Applied rewrites 81.0%

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

   12. Taylor expanded in alpha around 0

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

       1. lower-+.f64 68.5

          \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{2 + \beta}}{3 + \color{blue}{\beta}} \]

   14. Applied rewrites 68.5%

       \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{2 + \beta}}{\color{blue}{3 + \beta}} \]

   if 4e16 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

      1. metadata-eval 87.5

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

      2. metadata-eval 87.5

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   7. Applied rewrites 87.5%

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

Alternative 7: 98.7% accurate, 1.8× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 4e+16)
     (/
      (+ 1.0 beta)
      (* (+ 2.0 beta) (fma 3.0 (+ 2.0 beta) (* beta (+ 2.0 beta)))))
     (/ (/ (+ 1.0 alpha) t_0) (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * fma(3.0, (2.0 + beta), (beta * (2.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 4e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * fma(3.0, Float64(2.0 + beta), Float64(beta * Float64(2.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 * N[(2.0 + beta), $MachinePrecision] + N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 68.5

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

   9. Applied rewrites 68.5%

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

   if 4e16 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

      1. metadata-eval 87.5

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

      2. metadata-eval 87.5

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   7. Applied rewrites 87.5%

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

Alternative 8: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(3, 2 + \beta, \beta \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+16)
   (/
    (+ 1.0 beta)
    (* (+ 2.0 beta) (fma 3.0 (+ 2.0 beta) (* beta (+ 2.0 beta)))))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+16) {
		tmp = (1.0 + beta) / ((2.0 + beta) * fma(3.0, (2.0 + beta), (beta * (2.0 + beta))));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * fma(3.0, Float64(2.0 + beta), Float64(beta * Float64(2.0 + beta)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 * N[(2.0 + beta), $MachinePrecision] + N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e16

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 68.5

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

   9. Applied rewrites 68.5%

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

   if 1.5e16 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 87.0

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

   7. Applied rewrites 87.0%

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

Alternative 9: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \mathsf{fma}\left(3, 2 + \alpha, \alpha \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/
    (+ 1.0 alpha)
    (* (+ 2.0 alpha) (fma 3.0 (+ 2.0 alpha) (* alpha (+ 2.0 alpha)))))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = (1.0 + alpha) / ((2.0 + alpha) * fma(3.0, (2.0 + alpha), (alpha * (2.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + alpha) * fma(3.0, Float64(2.0 + alpha), Float64(alpha * Float64(2.0 + alpha)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 * N[(2.0 + alpha), $MachinePrecision] + N[(alpha * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if 6.20000000000000018 < beta

   1. Initial program 75.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 86.5

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

   7. Applied rewrites 86.5%

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

Alternative 10: 62.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1e+16)
   (/ (+ 1.0 alpha) (* (+ 2.0 beta) (+ 3.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1e+16) {
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.1d+16) then
        tmp = (1.0d0 + alpha) / ((2.0d0 + beta) * (3.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1e+16) {
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.1e+16:
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1e+16)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + beta) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.1e+16)
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.1e+16], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1e16

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around inf

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

      1. lift-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-+.f64 14.8

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

   10. Applied rewrites 14.8%

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

   if 3.1e16 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 87.0

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

   7. Applied rewrites 87.0%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.1% accurate, 2.6× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+15)
   (/ (+ 1.0 alpha) (* (+ 2.0 beta) (+ 3.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+15) {
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2d+15) then
        tmp = (1.0d0 + alpha) / ((2.0d0 + beta) * (3.0d0 + beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+15) {
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2e+15:
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta))
	else:
		tmp = ((1.0 + alpha) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+15)
		tmp = Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + beta) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2e+15)
		tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
	else
		tmp = ((1.0 + alpha) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+15], N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e15

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around inf

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

      1. lift-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-+.f64 14.8

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

   10. Applied rewrites 14.8%

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

   if 2e15 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 87.0

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 86.8%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1 + \alpha}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + alpha) / ((2.0d0 + beta) * (3.0d0 + beta))
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + beta) * Float64(3.0 + beta)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) / ((2.0 + beta) * (3.0 + beta));
end

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift-/.f64 N/A

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

4. Applied rewrites 90.2%

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

5. Taylor expanded in beta around inf

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

   1. lift-+.f64 50.1

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in alpha around 0

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

   1. lower-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lower-+.f64 35.5

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

10. Applied rewrites 35.5%

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

11. Final simplification 35.5%

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

Alternative 13: 52.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 28.6

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

5. Applied rewrites 28.6%

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

Alternative 14: 49.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 28.6

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

5. Applied rewrites 28.6%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 28.2%

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

Reproduce

herbie shell --seed 2025040 
(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)))