Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.6%

Time: 4.8s

Alternatives: 16

Speedup: 2.2×

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: 94.1% 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, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) - -2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_1}}{t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) - \left(\alpha + 1\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (- (+ beta alpha) -2.0)))
   (if (<= beta 2e+142)
     (/ (/ (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_1) t_1) t_0)
     (/
      (/ (- (+ alpha 1.0) (* (+ alpha 1.0) (/ (fma 2.0 alpha 4.0) beta))) beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) - -2.0;
	double tmp;
	if (beta <= 2e+142) {
		tmp = (((1.0 + fma(beta, alpha, (beta + alpha))) / t_1) / t_1) / t_0;
	} else {
		tmp = (((alpha + 1.0) - ((alpha + 1.0) * (fma(2.0, alpha, 4.0) / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) - -2.0)
	tmp = 0.0
	if (beta <= 2e+142)
		tmp = Float64(Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_1) / t_1) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) - Float64(Float64(alpha + 1.0) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+142], N[(N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] - N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e142

   1. Initial program 98.9%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.9%

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

   if 2.0000000000000001e142 < beta

   1. Initial program 81.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 89.6%

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

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 89.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) - -2\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_1}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) - \left(\alpha + 1\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (- (+ beta alpha) -2.0)))
   (if (<= beta 4e+151)
     (/ (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_1) (* t_1 t_0))
     (/
      (/ (- (+ alpha 1.0) (* (+ alpha 1.0) (/ (fma 2.0 alpha 4.0) beta))) beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) - -2.0;
	double tmp;
	if (beta <= 4e+151) {
		tmp = ((1.0 + fma(beta, alpha, (beta + alpha))) / t_1) / (t_1 * t_0);
	} else {
		tmp = (((alpha + 1.0) - ((alpha + 1.0) * (fma(2.0, alpha, 4.0) / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) - -2.0)
	tmp = 0.0
	if (beta <= 4e+151)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_1) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) - Float64(Float64(alpha + 1.0) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+151], N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] - N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000007e151

   1. Initial program 98.9%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   if 4.00000000000000007e151 < beta

   1. Initial program 80.3%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 80.3%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 89.1%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) - -2\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_0}}{t\_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{3 + \alpha \cdot \left(\frac{\beta}{\alpha} + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (+ beta alpha) -2.0)))
   (if (<= beta 5.5e+153)
     (/
      (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_0)
      (* t_0 (+ 3.0 (+ beta alpha))))
     (/ (/ (+ alpha 1.0) beta) (+ 3.0 (* alpha (+ (/ beta alpha) 1.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) - -2.0;
	double tmp;
	if (beta <= 5.5e+153) {
		tmp = ((1.0 + fma(beta, alpha, (beta + alpha))) / t_0) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (3.0 + (alpha * ((beta / alpha) + 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 <= 5.5e+153)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_0) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(3.0 + Float64(alpha * Float64(Float64(beta / alpha) + 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, 5.5e+153], N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha * N[(N[(beta / alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5000000000000003e153

   1. Initial program 98.4%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

   if 5.5000000000000003e153 < beta

   1. Initial program 81.7%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 91.2

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

   10. Applied rewrites 91.2%

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

Alternative 4: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{2 + \beta}}{\left(\beta + \alpha\right) - -2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) - -2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3e+16)
   (/
    (/ (/ (+ beta 1.0) (+ 2.0 beta)) (- (+ beta alpha) -2.0))
    (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) (- (+ alpha beta) -2.0)) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3e+16) {
		tmp = (((beta + 1.0) / (2.0 + beta)) / ((beta + alpha) - -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d+16) then
        tmp = (((beta + 1.0d0) / (2.0d0 + beta)) / ((beta + alpha) - (-2.0d0))) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / ((alpha + beta) - (-2.0d0))) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3e+16) {
		tmp = (((beta + 1.0) / (2.0 + beta)) / ((beta + alpha) - -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.3e+16:
		tmp = (((beta + 1.0) / (2.0 + beta)) / ((beta + alpha) - -2.0)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3e+16)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(2.0 + beta)) / Float64(Float64(beta + alpha) - -2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + beta) - -2.0)) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3e+16)
		tmp = (((beta + 1.0) / (2.0 + beta)) / ((beta + alpha) - -2.0)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.3e+16], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3e16

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if 2.3e16 < beta

   1. Initial program 86.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 82.6%

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

Alternative 5: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.25:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{2 + \alpha}}{\alpha - -2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) - -2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.25)
   (/
    (/ (/ (+ alpha 1.0) (+ 2.0 alpha)) (- alpha -2.0))
    (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) (- (+ alpha beta) -2.0)) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.25) {
		tmp = (((alpha + 1.0) / (2.0 + alpha)) / (alpha - -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.25d0) then
        tmp = (((alpha + 1.0d0) / (2.0d0 + alpha)) / (alpha - (-2.0d0))) / (3.0d0 + (beta + alpha))
    else
        tmp = ((alpha + 1.0d0) / ((alpha + beta) - (-2.0d0))) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.25) {
		tmp = (((alpha + 1.0) / (2.0 + alpha)) / (alpha - -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.25:
		tmp = (((alpha + 1.0) / (2.0 + alpha)) / (alpha - -2.0)) / (3.0 + (beta + alpha))
	else:
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.25)
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(2.0 + alpha)) / Float64(alpha - -2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + beta) - -2.0)) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.25)
		tmp = (((alpha + 1.0) / (2.0 + alpha)) / (alpha - -2.0)) / (3.0 + (beta + alpha));
	else
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.25], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.0%

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

   if 2.25 < beta

   1. Initial program 87.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

Alternative 6: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.25:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(\alpha - -2\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) - -2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.25)
   (/
    (/ (+ 1.0 alpha) (+ 2.0 alpha))
    (* (- alpha -2.0) (+ 3.0 (+ beta alpha))))
   (/ (/ (+ alpha 1.0) (- (+ alpha beta) -2.0)) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.25) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((alpha - -2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.25d0) then
        tmp = ((1.0d0 + alpha) / (2.0d0 + alpha)) / ((alpha - (-2.0d0)) * (3.0d0 + (beta + alpha)))
    else
        tmp = ((alpha + 1.0d0) / ((alpha + beta) - (-2.0d0))) / ((alpha + beta) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.25) {
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((alpha - -2.0) * (3.0 + (beta + alpha)));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.25:
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((alpha - -2.0) * (3.0 + (beta + alpha)))
	else:
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.25)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(Float64(alpha - -2.0) * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + beta) - -2.0)) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.25)
		tmp = ((1.0 + alpha) / (2.0 + alpha)) / ((alpha - -2.0) * (3.0 + (beta + alpha)));
	else
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.25], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha - -2.0), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.0%

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

       1. metadata-eval N/A

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

       2. associate-/l/ N/A

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

       3. lower-/.f64 N/A

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

   12. Applied rewrites 98.0%

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

   if 2.25 < beta

   1. Initial program 87.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

Alternative 7: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\alpha - -2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) - -2}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (/ (/ (fma 0.25 alpha 0.5) (- alpha -2.0)) (+ 3.0 (+ beta alpha)))
   (/ (/ (+ alpha 1.0) (- (+ alpha beta) -2.0)) (+ (+ alpha beta) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (fma(0.25, alpha, 0.5) / (alpha - -2.0)) / (3.0 + (beta + alpha));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + beta) - -2.0)) / ((alpha + beta) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(fma(0.25, alpha, 0.5) / Float64(alpha - -2.0)) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + beta) - -2.0)) / Float64(Float64(alpha + beta) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(N[(0.25 * alpha + 0.5), $MachinePrecision] / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in alpha around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 70.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\alpha - -2}}{3 + \left(\beta + \alpha\right)} \]

   13. Applied rewrites 70.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.25, \color{blue}{\alpha}, 0.5\right)}{\alpha - -2}}{3 + \left(\beta + \alpha\right)} \]

   if 2 < beta

   1. Initial program 87.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

Alternative 8: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\alpha - -2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= beta 4.5)
     (/ (/ (fma 0.25 alpha 0.5) (- alpha -2.0)) t_0)
     (/ (/ (+ alpha 1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 4.5) {
		tmp = (fma(0.25, alpha, 0.5) / (alpha - -2.0)) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(fma(0.25, alpha, 0.5) / Float64(alpha - -2.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.5], N[(N[(N[(0.25 * alpha + 0.5), $MachinePrecision] / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in alpha around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 70.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\alpha - -2}}{3 + \left(\beta + \alpha\right)} \]

   13. Applied rewrites 70.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.25, \color{blue}{\alpha}, 0.5\right)}{\alpha - -2}}{3 + \left(\beta + \alpha\right)} \]

   if 4.5 < beta

   1. Initial program 87.2%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 80.4

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

   7. Applied rewrites 80.4%

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

Alternative 9: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{\frac{0.5}{\alpha - -2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= beta 7.5)
     (/ (/ 0.5 (- alpha -2.0)) t_0)
     (/ (/ (+ alpha 1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.5) {
		tmp = (0.5 / (alpha - -2.0)) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / t_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 = 3.0d0 + (beta + alpha)
    if (beta <= 7.5d0) then
        tmp = (0.5d0 / (alpha - (-2.0d0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 7.5) {
		tmp = (0.5 / (alpha - -2.0)) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (beta + alpha)
	tmp = 0
	if beta <= 7.5:
		tmp = (0.5 / (alpha - -2.0)) / t_0
	else:
		tmp = ((alpha + 1.0) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(Float64(0.5 / Float64(alpha - -2.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 7.5)
		tmp = (0.5 / (alpha - -2.0)) / t_0;
	else
		tmp = ((alpha + 1.0) / beta) / t_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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.5], N[(N[(0.5 / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   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. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 85.7%

       \[\leadsto \frac{\frac{0.5}{\alpha - -2}}{3 + \left(\beta + \alpha\right)} \]

   if 7.5 < beta

   1. Initial program 87.2%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 80.4

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

   7. Applied rewrites 80.4%

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

Alternative 10: 61.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + \beta\right) - -2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.86e+154)
   (/ (+ alpha 1.0) (* (- (+ alpha beta) -2.0) (+ (+ alpha beta) 3.0)))
   (/ (/ alpha beta) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.86e+154) {
		tmp = (alpha + 1.0) / (((alpha + beta) - -2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = (alpha / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.86d+154) then
        tmp = (alpha + 1.0d0) / (((alpha + beta) - (-2.0d0)) * ((alpha + beta) + 3.0d0))
    else
        tmp = (alpha / beta) / (3.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.86e+154) {
		tmp = (alpha + 1.0) / (((alpha + beta) - -2.0) * ((alpha + beta) + 3.0));
	} else {
		tmp = (alpha / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.86e+154:
		tmp = (alpha + 1.0) / (((alpha + beta) - -2.0) * ((alpha + beta) + 3.0))
	else:
		tmp = (alpha / beta) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.86e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(Float64(alpha + beta) - -2.0) * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.86e+154)
		tmp = (alpha + 1.0) / (((alpha + beta) - -2.0) * ((alpha + beta) + 3.0));
	else
		tmp = (alpha / beta) / (3.0 + (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.86e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.86000000000000014e154

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 39.2%

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

   if 1.86000000000000014e154 < beta

   1. Initial program 81.3%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in alpha around inf

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

      1. +-commutative 91.0

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

   10. Applied rewrites 91.0%

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

Alternative 11: 55.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.86e+154)
   (/ (+ 1.0 alpha) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.86e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.86d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.86e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.86e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.86e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.86e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.86e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.86000000000000014e154

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 17.3

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

   5. Applied rewrites 17.3%

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

   if 1.86000000000000014e154 < beta

   1. Initial program 81.3%

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

   3. Taylor expanded in 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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 91.5%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 90.9

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

      4. +-commutative 90.9

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

   10. Applied rewrites 90.9%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha + 1.0) / 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 = ((alpha + 1.0d0) / beta) / (3.0d0 + beta)
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / 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[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 95.6%

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

5. Taylor expanded in beta around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

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

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

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

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 28.9

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

7. Applied rewrites 28.9%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 28.8%

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

Alternative 13: 55.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   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 29.4

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

5. Applied rewrites 29.4%

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

   1. associate-/r* N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 29.3

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

7. Applied rewrites 29.3%

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

Alternative 14: 52.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      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 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 34.7%

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

   if 1 < alpha

   1. Initial program 86.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 18.0%

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

Alternative 15: 52.9% 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 95.6%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   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 29.4

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

5. Applied rewrites 29.4%

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

Alternative 16: 50.1% 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 95.6%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   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 29.4

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

5. Applied rewrites 29.4%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 28.6%

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

Reproduce

herbie shell --seed 2025044 
(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)))