Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.3%

Time: 36.1s

Alternatives: 7

Speedup: N/A×

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: 94.4% 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: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\\ t_1 := \frac{1 + \beta}{2 + \beta}\\ t_2 := \left(\alpha + \beta\right) + 2\\ t_3 := \frac{1 + \beta}{{\left(2 + \beta\right)}^{3}}\\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_3 - \frac{1 + \beta}{{\left(2 + \beta\right)}^{4}}, \alpha, t\_3\right) - t\_0, \alpha, t\_1\right) - t\_0, \alpha, t\_1\right)}{t\_2}}{t\_2 + 1} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 2.0)))
        (t_1 (/ (+ 1.0 beta) (+ 2.0 beta)))
        (t_2 (+ (+ alpha beta) 2.0))
        (t_3 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 3.0))))
   (/
    (/
     (fma
      (-
       (fma
        (- (fma (- t_3 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 4.0))) alpha t_3) t_0)
        alpha
        t_1)
       t_0)
      alpha
      t_1)
     t_2)
    (+ t_2 1.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + beta) / pow((2.0 + beta), 2.0);
	double t_1 = (1.0 + beta) / (2.0 + beta);
	double t_2 = (alpha + beta) + 2.0;
	double t_3 = (1.0 + beta) / pow((2.0 + beta), 3.0);
	return (fma((fma((fma((t_3 - ((1.0 + beta) / pow((2.0 + beta), 4.0))), alpha, t_3) - t_0), alpha, t_1) - t_0), alpha, t_1) / t_2) / (t_2 + 1.0);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 2.0))
	t_1 = Float64(Float64(1.0 + beta) / Float64(2.0 + beta))
	t_2 = Float64(Float64(alpha + beta) + 2.0)
	t_3 = Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 3.0))
	return Float64(Float64(fma(Float64(fma(Float64(fma(Float64(t_3 - Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 4.0))), alpha, t_3) - t_0), alpha, t_1) - t_0), alpha, t_1) / t_2) / Float64(t_2 + 1.0))
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + beta), $MachinePrecision] / N[Power[N[(2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + beta), $MachinePrecision] / N[Power[N[(2.0 + beta), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(t$95$3 - N[(N[(1.0 + beta), $MachinePrecision] / N[Power[N[(2.0 + beta), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * alpha + t$95$3), $MachinePrecision] - t$95$0), $MachinePrecision] * alpha + t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] * alpha + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 75.5%

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

6. Final simplification 75.5%

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

Alternative 2: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\\ t_1 := \frac{1 + \beta}{2 + \beta}\\ t_2 := \frac{1 + \beta}{{\left(2 + \beta\right)}^{3}}\\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2 - \frac{1 + \beta}{{\left(2 + \beta\right)}^{4}}, \alpha, t\_2\right) - t\_0, \alpha, t\_1\right) - t\_0, \alpha, t\_1\right)}{\left(\alpha + \beta\right) + 2}}{\beta \cdot \left(\frac{3 + \alpha}{\beta} - -1\right)} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 2.0)))
        (t_1 (/ (+ 1.0 beta) (+ 2.0 beta)))
        (t_2 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 3.0))))
   (/
    (/
     (fma
      (-
       (fma
        (- (fma (- t_2 (/ (+ 1.0 beta) (pow (+ 2.0 beta) 4.0))) alpha t_2) t_0)
        alpha
        t_1)
       t_0)
      alpha
      t_1)
     (+ (+ alpha beta) 2.0))
    (* beta (- (/ (+ 3.0 alpha) beta) -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + beta) / pow((2.0 + beta), 2.0);
	double t_1 = (1.0 + beta) / (2.0 + beta);
	double t_2 = (1.0 + beta) / pow((2.0 + beta), 3.0);
	return (fma((fma((fma((t_2 - ((1.0 + beta) / pow((2.0 + beta), 4.0))), alpha, t_2) - t_0), alpha, t_1) - t_0), alpha, t_1) / ((alpha + beta) + 2.0)) / (beta * (((3.0 + alpha) / beta) - -1.0));
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 2.0))
	t_1 = Float64(Float64(1.0 + beta) / Float64(2.0 + beta))
	t_2 = Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 3.0))
	return Float64(Float64(fma(Float64(fma(Float64(fma(Float64(t_2 - Float64(Float64(1.0 + beta) / (Float64(2.0 + beta) ^ 4.0))), alpha, t_2) - t_0), alpha, t_1) - t_0), alpha, t_1) / Float64(Float64(alpha + beta) + 2.0)) / Float64(beta * Float64(Float64(Float64(3.0 + alpha) / beta) - -1.0)))
end

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 75.5%

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

6. Taylor expanded in beta around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 75.4

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

8. Applied rewrites 75.4%

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

9. Final simplification 75.4%

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

Alternative 3: 98.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := {\left(2 + \beta\right)}^{2}\\ t_1 := \frac{1 + \beta}{2 + \beta}\\ \mathbf{if}\;\beta \leq 190000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, {\beta}^{-1}, \frac{\alpha}{\beta}\right) - 1}{\beta}, \alpha, t\_1\right) - \frac{1 + \beta}{t\_0}, \alpha, t\_1\right)}{\beta \cdot \left(\frac{\alpha}{\beta} - -1\right) + 2}}{\left(\beta \cdot \left(1 + \frac{\alpha}{\beta}\right) + 2\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (pow (+ 2.0 beta) 2.0)) (t_1 (/ (+ 1.0 beta) (+ 2.0 beta))))
   (if (<= beta 190000.0)
     (/ (+ 1.0 beta) (* (+ 3.0 beta) t_0))
     (/
      (/
       (fma
        (-
         (fma
          (/ (- (fma 4.0 (pow beta -1.0) (/ alpha beta)) 1.0) beta)
          alpha
          t_1)
         (/ (+ 1.0 beta) t_0))
        alpha
        t_1)
       (+ (* beta (- (/ alpha beta) -1.0)) 2.0))
      (+ (+ (* beta (+ 1.0 (/ alpha beta))) 2.0) 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = pow((2.0 + beta), 2.0);
	double t_1 = (1.0 + beta) / (2.0 + beta);
	double tmp;
	if (beta <= 190000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * t_0);
	} else {
		tmp = (fma((fma(((fma(4.0, pow(beta, -1.0), (alpha / beta)) - 1.0) / beta), alpha, t_1) - ((1.0 + beta) / t_0)), alpha, t_1) / ((beta * ((alpha / beta) - -1.0)) + 2.0)) / (((beta * (1.0 + (alpha / beta))) + 2.0) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + beta) ^ 2.0
	t_1 = Float64(Float64(1.0 + beta) / Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 190000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * t_0));
	else
		tmp = Float64(Float64(fma(Float64(fma(Float64(Float64(fma(4.0, (beta ^ -1.0), Float64(alpha / beta)) - 1.0) / beta), alpha, t_1) - Float64(Float64(1.0 + beta) / t_0)), alpha, t_1) / Float64(Float64(beta * Float64(Float64(alpha / beta) - -1.0)) + 2.0)) / Float64(Float64(Float64(beta * Float64(1.0 + Float64(alpha / beta))) + 2.0) + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[Power[N[(2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 190000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(4.0 * N[Power[beta, -1.0], $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $MachinePrecision] * alpha + t$95$1), $MachinePrecision] - N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * alpha + t$95$1), $MachinePrecision] / N[(N[(beta * N[(N[(alpha / beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta * N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9e5

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-+.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if 1.9e5 < beta

   1. Initial program 83.6%

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

   3. Taylor expanded in alpha around 0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 84.2

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in beta around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 84.2

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

   11. Applied rewrites 84.2%

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

   12. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 84.2

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

   14. Applied rewrites 84.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 190000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(4, {\beta}^{-1}, \frac{\alpha}{\beta}\right) - 1}{\beta}, \alpha, \frac{1 + \beta}{2 + \beta}\right) - \frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}, \alpha, \frac{1 + \beta}{2 + \beta}\right)}{\beta \cdot \left(\frac{\alpha}{\beta} - -1\right) + 2}}{\left(\beta \cdot \left(1 + \frac{\alpha}{\beta}\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 31000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 31000000.0)
     (/ (+ 1.0 beta) (* (+ 3.0 beta) (pow (+ 2.0 beta) 2.0)))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       t_0)
      (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 31000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * pow((2.0 + beta), 2.0));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 31000000.0d0) then
        tmp = (1.0d0 + beta) / ((3.0d0 + beta) * ((2.0d0 + beta) ** 2.0d0))
    else
        tmp = ((((((1.0d0 + alpha) / beta) + alpha) + 1.0d0) - ((1.0d0 + alpha) * ((2.0d0 + alpha) / beta))) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 31000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * Math.pow((2.0 + beta), 2.0));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 31000000.0:
		tmp = (1.0 + beta) / ((3.0 + beta) * math.pow((2.0 + beta), 2.0))
	else:
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / (t_0 + 1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 31000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * (Float64(2.0 + beta) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 31000000.0)
		tmp = (1.0 + beta) / ((3.0 + beta) * ((2.0 + beta) ^ 2.0));
	else
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 31000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[Power[N[(2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1e7

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-+.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if 3.1e7 < beta

   1. Initial program 83.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. div-add N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. div-add N/A

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

      18. lower-/.f64 N/A

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

   5. Applied rewrites 84.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 31000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 130000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \frac{6}{{\beta}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\alpha \cdot \left(1 + \frac{\beta}{\alpha}\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 130000000.0)
   (/ (+ 1.0 beta) (* (+ 3.0 beta) (pow (+ 2.0 beta) 2.0)))
   (if (<= beta 3.2e+157)
     (-
      (fma
       alpha
       (-
        (fma -3.0 (/ alpha (pow beta 3.0)) (pow beta -2.0))
        (* 9.0 (pow beta -3.0)))
       (pow beta -2.0))
      (/ 6.0 (pow beta 3.0)))
     (/
      (/
       (-
        (+ (+ (/ (+ 1.0 alpha) beta) alpha) 1.0)
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       (+ (* alpha (+ 1.0 (/ beta alpha))) 2.0))
      (+ (+ (+ alpha beta) 2.0) 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 130000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * pow((2.0 + beta), 2.0));
	} else if (beta <= 3.2e+157) {
		tmp = fma(alpha, (fma(-3.0, (alpha / pow(beta, 3.0)), pow(beta, -2.0)) - (9.0 * pow(beta, -3.0))), pow(beta, -2.0)) - (6.0 / pow(beta, 3.0));
	} else {
		tmp = ((((((1.0 + alpha) / beta) + alpha) + 1.0) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / ((alpha * (1.0 + (beta / alpha))) + 2.0)) / (((alpha + beta) + 2.0) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 130000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * (Float64(2.0 + beta) ^ 2.0)));
	elseif (beta <= 3.2e+157)
		tmp = Float64(fma(alpha, Float64(fma(-3.0, Float64(alpha / (beta ^ 3.0)), (beta ^ -2.0)) - Float64(9.0 * (beta ^ -3.0))), (beta ^ -2.0)) - Float64(6.0 / (beta ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + alpha) / beta) + alpha) + 1.0) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / Float64(Float64(alpha * Float64(1.0 + Float64(beta / alpha))) + 2.0)) / Float64(Float64(Float64(alpha + beta) + 2.0) + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 130000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[Power[N[(2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.2e+157], N[(N[(alpha * N[(N[(-3.0 * N[(alpha / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(9.0 * N[Power[beta, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(6.0 / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha * N[(1.0 + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.3e8

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-+.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if 1.3e8 < beta < 3.1999999999999999e157

   1. Initial program 92.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(7 + \left(\alpha + 2 \cdot \alpha\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(3, \alpha, 7\right)}{\beta}}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \frac{6}{\color{blue}{{\beta}^{3}}} \]

   8. Applied rewrites 72.5%

      \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]

   if 3.1999999999999999e157 < beta

   1. Initial program 74.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. div-add N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. div-add N/A

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

      18. lower-/.f64 N/A

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 130000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \frac{6}{{\beta}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{\alpha \cdot \left(1 + \frac{\beta}{\alpha}\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 130000000:\\ \;\;\;\;\frac{1 + \beta}{\left(3 + \beta\right) \cdot {\left(2 + \beta\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \frac{6}{{\beta}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 130000000.0)
   (/ (+ 1.0 beta) (* (+ 3.0 beta) (pow (+ 2.0 beta) 2.0)))
   (-
    (fma
     alpha
     (-
      (fma -3.0 (/ alpha (pow beta 3.0)) (pow beta -2.0))
      (* 9.0 (pow beta -3.0)))
     (pow beta -2.0))
    (/ 6.0 (pow beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 130000000.0) {
		tmp = (1.0 + beta) / ((3.0 + beta) * pow((2.0 + beta), 2.0));
	} else {
		tmp = fma(alpha, (fma(-3.0, (alpha / pow(beta, 3.0)), pow(beta, -2.0)) - (9.0 * pow(beta, -3.0))), pow(beta, -2.0)) - (6.0 / pow(beta, 3.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 130000000.0)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * (Float64(2.0 + beta) ^ 2.0)));
	else
		tmp = Float64(fma(alpha, Float64(fma(-3.0, Float64(alpha / (beta ^ 3.0)), (beta ^ -2.0)) - Float64(9.0 * (beta ^ -3.0))), (beta ^ -2.0)) - Float64(6.0 / (beta ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 130000000.0], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[Power[N[(2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha * N[(N[(-3.0 * N[(alpha / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(9.0 * N[Power[beta, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(6.0 / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e8

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-+.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if 1.3e8 < beta

   1. Initial program 83.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(7 + \left(\alpha + 2 \cdot \alpha\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(3, \alpha, 7\right)}{\beta}}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \frac{6}{\color{blue}{{\beta}^{3}}} \]

   8. Applied rewrites 80.6%

      \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \frac{6}{{\beta}^{3}} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (-
  (fma
   alpha
   (-
    (fma -3.0 (/ alpha (pow beta 3.0)) (pow beta -2.0))
    (* 9.0 (pow beta -3.0)))
   (pow beta -2.0))
  (/ 6.0 (pow beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(alpha, (fma(-3.0, (alpha / pow(beta, 3.0)), pow(beta, -2.0)) - (9.0 * pow(beta, -3.0))), pow(beta, -2.0)) - (6.0 / pow(beta, 3.0));
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(fma(alpha, Float64(fma(-3.0, Float64(alpha / (beta ^ 3.0)), (beta ^ -2.0)) - Float64(9.0 * (beta ^ -3.0))), (beta ^ -2.0)) - Float64(6.0 / (beta ^ 3.0)))
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(alpha * N[(N[(-3.0 * N[(alpha / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(9.0 * N[Power[beta, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[beta, -2.0], $MachinePrecision]), $MachinePrecision] - N[(6.0 / N[Power[beta, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(1 + \left(\alpha + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(7 + \left(\alpha + 2 \cdot \alpha\right)\right)}{\beta}}{\color{blue}{{\beta}^{2}}} \]

5. Applied rewrites 27.4%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(3, \alpha, 7\right)}{\beta}}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\alpha \cdot \left(\left(-3 \cdot \frac{\alpha}{{\beta}^{3}} + \frac{1}{{\beta}^{2}}\right) - 9 \cdot \frac{1}{{\beta}^{3}}\right) + \frac{1}{{\beta}^{2}}\right) - \frac{6}{\color{blue}{{\beta}^{3}}} \]

8. Applied rewrites 27.8%

   \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(-3, \frac{\alpha}{{\beta}^{3}}, {\beta}^{-2}\right) - 9 \cdot {\beta}^{-3}, {\beta}^{-2}\right) - \color{blue}{\frac{6}{{\beta}^{3}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))