Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.7%

Time: 6.3s

Alternatives: 15

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_1}}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1 - \alpha, \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}, \left(\frac{\alpha - -1}{\beta} + \alpha\right) - -1\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.5e+151)
     (/ (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_1) (* t_0 t_1))
     (/
      (/
       (fma
        (- -1.0 alpha)
        (/ (fma 2.0 alpha 4.0) beta)
        (- (+ (/ (- alpha -1.0) beta) alpha) -1.0))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.5e+151) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_1) / (t_0 * t_1);
	} else {
		tmp = (fma((-1.0 - alpha), (fma(2.0, alpha, 4.0) / beta), ((((alpha - -1.0) / beta) + 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))
	t_1 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.5e+151)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_1) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(fma(Float64(-1.0 - alpha), Float64(fma(2.0, alpha, 4.0) / beta), Float64(Float64(Float64(Float64(alpha - -1.0) / beta) + 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]}, Block[{t$95$1 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+151], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision] + N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e151

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 96.1%

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

   if 1.5e151 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 97.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.5e+151)
     (/
      (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      (+ 3.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.5e+151) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (3.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.5e+151)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+151], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e151

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 96.1%

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

   if 1.5e151 < beta

   1. Initial program 74.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-+.f64 97.1

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

   8. Applied rewrites 97.1%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.5e+151)
     (/
      (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ 3.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.5e+151) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.5e+151)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(3.0 + beta) + alpha));
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+151], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e151

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 96.1%

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

   if 1.5e151 < beta

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 74.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 74.8

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 97.2

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

   7. Applied rewrites 97.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 97.2

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-+l+ N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.2

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

   9. Applied rewrites 97.2%

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

   11. Applied rewrites 97.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{2 + \left(\beta + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 4e+77)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ 3.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 4e+77) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4e+77)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(3.0 + beta) + alpha));
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+77], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999993e77

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 92.6%

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

   if 3.99999999999999993e77 < beta

   1. Initial program 79.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 79.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 89.6

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-+l+ N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 89.6

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

   9. Applied rewrites 89.6%

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

   11. Applied rewrites 89.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0 \cdot \left(\left(3 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.9e+17)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (* t_0 (* (+ 3.0 beta) (+ 2.0 beta))))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ 3.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.9e+17) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (t_0 * ((3.0 + beta) * (2.0 + beta)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.9e+17)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(t_0 * Float64(Float64(3.0 + beta) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(3.0 + beta) + alpha));
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.9e+17], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9e17

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if 1.9e17 < beta

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 81.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 82.7

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

   7. Applied rewrites 82.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 82.7

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-+l+ N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 82.7

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

   9. Applied rewrites 82.7%

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

   11. Applied rewrites 82.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\left(3 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta - -1}{t\_0 \cdot \left(\left(3 + \beta\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 5e+16)
     (/ (- beta -1.0) (* t_0 (* (+ 3.0 beta) t_0)))
     (/ (/ (+ 1.0 alpha) t_0) (+ (+ 3.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 5d+16) then
        tmp = (beta - (-1.0d0)) / (t_0 * ((3.0d0 + beta) * t_0))
    else
        tmp = ((1.0d0 + alpha) / t_0) / ((3.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	} else {
		tmp = ((1.0 + alpha) / t_0) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5e+16:
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0))
	else:
		tmp = ((1.0 + alpha) / t_0) / ((3.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5e+16)
		tmp = Float64(Float64(beta - -1.0) / Float64(t_0 * Float64(Float64(3.0 + beta) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(3.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 5e+16)
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	else
		tmp = ((1.0 + alpha) / t_0) / ((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_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+16], N[(N[(beta - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + beta), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 87.0

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

   10. Applied rewrites 87.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. metadata-eval N/A

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

       7. associate-/l/ N/A

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

   12. Applied rewrites 87.1%

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

   if 5e16 < beta

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 81.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 82.7

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

   7. Applied rewrites 82.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 82.7

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-+l+ N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 82.7

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

   9. Applied rewrites 82.7%

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

   11. Applied rewrites 82.7%

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

4. Final simplification 85.4%

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

Alternative 7: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta - -1}{t\_0 \cdot \left(\left(3 + \beta\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 5e+16)
     (/ (- beta -1.0) (* t_0 (* (+ 3.0 beta) t_0)))
     (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	} else {
		tmp = ((1.0 + 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 5d+16) then
        tmp = (beta - (-1.0d0)) / (t_0 * ((3.0d0 + beta) * t_0))
    else
        tmp = ((1.0d0 + 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 t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5e+16) {
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5e+16:
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5e+16)
		tmp = Float64(Float64(beta - -1.0) / Float64(t_0 * Float64(Float64(3.0 + beta) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + 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)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 5e+16)
		tmp = (beta - -1.0) / (t_0 * ((3.0 + beta) * t_0));
	else
		tmp = ((1.0 + 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_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+16], N[(N[(beta - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + beta), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 87.0

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

   10. Applied rewrites 87.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. metadata-eval N/A

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

       7. associate-/l/ N/A

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

   12. Applied rewrites 87.1%

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

   if 5e16 < beta

   1. Initial program 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 81.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 71.4

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

   7. Applied rewrites 71.4%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.0

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

   10. Applied rewrites 82.0%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 82.2

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

   13. Applied rewrites 82.2%

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

4. Final simplification 85.2%

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

Alternative 8: 62.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{1 + \alpha}{t\_0 \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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 1.5e+151)
     (/ (+ 1.0 alpha) (* t_0 (+ 2.0 (+ beta alpha))))
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.5e+151) {
		tmp = (1.0 + alpha) / (t_0 * (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 1.5d+151) then
        tmp = (1.0d0 + alpha) / (t_0 * (2.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / 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 <= 1.5e+151) {
		tmp = (1.0 + alpha) / (t_0 * (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 1.5e+151:
		tmp = (1.0 + alpha) / (t_0 * (2.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / 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 <= 1.5e+151)
		tmp = Float64(Float64(1.0 + alpha) / Float64(t_0 * Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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 <= 1.5e+151)
		tmp = (1.0 + alpha) / (t_0 * (2.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / 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, 1.5e+151], N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 * N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e151

   1. Initial program 97.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 97.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 28.4

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

   7. Applied rewrites 28.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 28.4

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-+l+ N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 28.4

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

   9. Applied rewrites 28.4%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. associate-+l+ N/A

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

       5. +-commutative N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. associate-+l+ N/A

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

       9. lift-/.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. +-commutative N/A

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

       12. lift-+.f64 N/A

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

       13. +-commutative N/A

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

   11. Applied rewrites 43.6%

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

   if 1.5e151 < beta

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2 N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 74.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 74.8

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 97.1

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

   10. Applied rewrites 97.1%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 97.1

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

   13. Applied rewrites 97.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{1 + \alpha}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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) / (3.0d0 + (beta + alpha))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2 N/A

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

   4. associate-+r+ N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. associate--l+ N/A

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

   9. lift-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f64 92.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 92.9

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

4. Applied rewrites 92.9%

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

5. Taylor expanded in alpha around 0

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

   1. lower-+.f64 70.3

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

7. Applied rewrites 70.3%

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

8. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 33.4

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

10. Applied rewrites 33.4%

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

11. Taylor expanded in alpha around 0

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

    1. lower-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 33.5

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

13. Applied rewrites 33.5%

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

14. Final simplification 33.5%

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

Alternative 10: 55.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.35e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.35000000000000003e154

   1. Initial program 97.0%

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

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 20.0

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

   5. Applied rewrites 20.0%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 74.3%

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

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 85.0%

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

   10. Applied rewrites 96.5%

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

Alternative 11: 55.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2 N/A

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

   4. associate-+r+ N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. associate--l+ N/A

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

   9. lift-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f64 92.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 92.9

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

4. Applied rewrites 92.9%

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

5. Taylor expanded in alpha around 0

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

   1. lower-+.f64 70.3

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

7. Applied rewrites 70.3%

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

8. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 33.4

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

10. Applied rewrites 33.4%

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

11. Final simplification 33.4%

    \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \beta} \]
12. Add Preprocessing

Alternative 12: 55.6% 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 92.9%

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

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 31.6

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

5. Applied rewrites 31.6%

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

7. Applied rewrites 33.8%

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

Alternative 13: 52.0% 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

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 38.4%

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

   if 1 < alpha

   1. Initial program 79.6%

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

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 17.4%

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

Alternative 14: 52.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 31.6

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

5. Applied rewrites 31.6%

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

Alternative 15: 32.2% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 31.6

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

5. Applied rewrites 31.6%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 19.0%

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

Reproduce

herbie shell --seed 2025017 
(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)))