Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 99.7%

Time: 6.8s

Alternatives: 10

Speedup: 75.4×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 16.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	return (1.0 / ((fma(2.0, i, ((alpha + beta) - -1.0)) / ((alpha + beta) + i)) * (((alpha + beta) / i) - -2.0))) * (fma((i / t_0), (i + (alpha + beta)), (alpha * (beta / t_0))) / (fma(2.0, i, (beta + alpha)) - 1.0));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	return Float64(Float64(1.0 / Float64(Float64(fma(2.0, i, Float64(Float64(alpha + beta) - -1.0)) / Float64(Float64(alpha + beta) + i)) * Float64(Float64(Float64(alpha + beta) / i) - -2.0))) * Float64(fma(Float64(i / t_0), Float64(i + Float64(alpha + beta)), Float64(alpha * Float64(beta / t_0))) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0)))
end

Wolfram

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

Derivation

1. Initial program 16.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr-1 N/A

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

3. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. div-flip N/A

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

   13. lift-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. add-to-fraction N/A

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

   18. lift-/.f64 N/A

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

   19. lift-+.f64 N/A

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - -1\right)}{\left(\alpha + \beta\right) + i} \cdot \left(\frac{\alpha + \beta}{i} - -2\right)}} \cdot \frac{\mathsf{fma}\left(\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, i + \left(\alpha + \beta\right), \alpha \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \]
8. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = i / t_0;
	return (t_1 * (((alpha + beta) + i) / fma(2.0, i, ((alpha + beta) - -1.0)))) * (fma(t_1, (i + (alpha + beta)), (alpha * (beta / t_0))) / (fma(2.0, i, (beta + alpha)) - 1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr-1 N/A

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

3. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.7

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.7

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

   19. lift--.f64 N/A

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - -1\right)}\right)} \cdot \frac{\mathsf{fma}\left(\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, i + \left(\alpha + \beta\right), \alpha \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \]
8. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	return ((((alpha + beta) + i) / (((alpha + beta) / i) - -2.0)) / fma(2.0, i, ((alpha + beta) - -1.0))) * (fma((i / t_0), (i + (alpha + beta)), (alpha * (beta / t_0))) / (fma(2.0, i, (beta + alpha)) - 1.0));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	return Float64(Float64(Float64(Float64(Float64(alpha + beta) + i) / Float64(Float64(Float64(alpha + beta) / i) - -2.0)) / fma(2.0, i, Float64(Float64(alpha + beta) - -1.0))) * Float64(fma(Float64(i / t_0), Float64(i + Float64(alpha + beta)), Float64(alpha * Float64(beta / t_0))) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0)))
end

Wolfram

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

Derivation

1. Initial program 16.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr-1 N/A

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

3. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\alpha + \beta}{i} - -2}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - -1\right)}} \cdot \frac{\mathsf{fma}\left(\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, i + \left(\alpha + \beta\right), \alpha \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (1.0 / ((fma(2.0, i, (beta - -1.0)) / (beta + i)) * ((beta / i) - -2.0))) * (fma((i / fma(2.0, i, beta)), (i + beta), (alpha * (beta / fma(2.0, i, beta)))) / (fma(2.0, i, (beta + alpha)) - 1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr-1 N/A

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

3. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. div-flip N/A

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

   13. lift-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. add-to-fraction N/A

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

   18. lift-/.f64 N/A

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

   19. lift-+.f64 N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in alpha around 0

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

13. Applied rewrites 99.7%

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

14. Taylor expanded in alpha around 0

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

16. Applied rewrites 99.5%

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

17. Taylor expanded in alpha around 0

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

19. Applied rewrites 99.5%

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

20. Taylor expanded in alpha around 0

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

22. Applied rewrites 99.5%

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

23. Taylor expanded in alpha around 0

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

25. Applied rewrites 99.5%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, i, \beta - -1\right)}{\beta + i} \cdot \left(\frac{\beta}{i} - -2\right)} \cdot \frac{\mathsf{fma}\left(\frac{i}{\mathsf{fma}\left(2, i, \beta\right)}, i + \beta, \alpha \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \color{blue}{\beta}\right)}\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \]
26. Add Preprocessing

Alternative 5: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 1.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{i \cdot \frac{\mathsf{fma}\left(i + \beta, i, \alpha \cdot \beta\right)}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - -1} \cdot \frac{\frac{i + \beta}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.2e+149)
   (*
    (/
     (* i (/ (fma (+ i beta) i (* alpha beta)) (fma 2.0 i beta)))
     (- (fma 2.0 i beta) -1.0))
    (/ (/ (+ i beta) (fma 2.0 i beta)) (- (fma 2.0 i beta) 1.0)))
   0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.2e+149) {
		tmp = ((i * (fma((i + beta), i, (alpha * beta)) / fma(2.0, i, beta))) / (fma(2.0, i, beta) - -1.0)) * (((i + beta) / fma(2.0, i, beta)) / (fma(2.0, i, beta) - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.2e+149)
		tmp = Float64(Float64(Float64(i * Float64(fma(Float64(i + beta), i, Float64(alpha * beta)) / fma(2.0, i, beta))) / Float64(fma(2.0, i, beta) - -1.0)) * Float64(Float64(Float64(i + beta) / fma(2.0, i, beta)) / Float64(fma(2.0, i, beta) - 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 1.2e+149], N[(N[(N[(i * N[(N[(N[(i + beta), $MachinePrecision] * i + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i + beta), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1.20000000000000006e149

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 43.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 43.7%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 43.7%

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

   15. Taylor expanded in alpha around 0

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

   17. Applied rewrites 43.7%

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

   18. Taylor expanded in alpha around 0

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

   20. Applied rewrites 43.7%

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

   21. Taylor expanded in alpha around 0

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

   23. Applied rewrites 43.7%

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

   if 1.20000000000000006e149 < i

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.5%

       \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 1.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \left(\beta + i\right) \cdot i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta\right), \mathsf{fma}\left(2, i, \beta\right), -1\right)}}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \frac{i + \beta}{2 + \frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.2e+149)
   (*
    (/
     (/
      (fma alpha beta (* (+ beta i) i))
      (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0))
     (fma 2.0 i beta))
    (/ (+ i beta) (+ 2.0 (/ beta i))))
   0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.2e+149) {
		tmp = ((fma(alpha, beta, ((beta + i) * i)) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)) / fma(2.0, i, beta)) * ((i + beta) / (2.0 + (beta / i)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.2e+149)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(Float64(beta + i) * i)) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)) / fma(2.0, i, beta)) * Float64(Float64(i + beta) / Float64(2.0 + Float64(beta / i))));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 1.2e+149], N[(N[(N[(N[(alpha * beta + N[(N[(beta + i), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(2.0 + N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1.20000000000000006e149

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

   5. Applied rewrites 24.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 38.5%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 38.5%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 38.5%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 38.5%

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

   20. Taylor expanded in alpha around 0

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

   22. Applied rewrites 38.5%

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

   23. Taylor expanded in alpha around 0

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

   25. Applied rewrites 38.5%

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

   if 1.20000000000000006e149 < i

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.5%

       \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+168}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - -1\right)}{\left(\alpha + \beta\right) + i} \cdot \left(\frac{\alpha + \beta}{i} - -2\right)} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.65e+168)
   0.0625
   (*
    (/
     1.0
     (*
      (/ (fma 2.0 i (- (+ alpha beta) -1.0)) (+ (+ alpha beta) i))
      (- (/ (+ alpha beta) i) -2.0)))
    (/ (+ alpha i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.65e+168) {
		tmp = 0.0625;
	} else {
		tmp = (1.0 / ((fma(2.0, i, ((alpha + beta) - -1.0)) / ((alpha + beta) + i)) * (((alpha + beta) / i) - -2.0))) * ((alpha + i) / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.65e+168)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(fma(2.0, i, Float64(Float64(alpha + beta) - -1.0)) / Float64(Float64(alpha + beta) + i)) * Float64(Float64(Float64(alpha + beta) / i) - -2.0))) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.65e+168], 0.0625, N[(N[(1.0 / N[(N[(N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999e168

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.5%

       \[\leadsto \color{blue}{0.0625} \]

   if 1.6499999999999999e168 < beta

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.6

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

   10. Applied rewrites 30.6%

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

Alternative 8: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\beta} \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{t\_0}}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= i 5e+35)
     (*
      (/ (* i (+ alpha i)) beta)
      (/ (/ (+ i (+ alpha beta)) t_0) (- t_0 1.0)))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (i <= 5e+35) {
		tmp = ((i * (alpha + i)) / beta) * (((i + (alpha + beta)) / t_0) / (t_0 - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (i <= 5e+35)
		tmp = Float64(Float64(Float64(i * Float64(alpha + i)) / beta) * Float64(Float64(Float64(i + Float64(alpha + beta)) / t_0) / Float64(t_0 - 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 5e+35], N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if i < 5.00000000000000021e35

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 22.8

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

   8. Applied rewrites 22.8%

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

   if 5.00000000000000021e35 < i

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.5%

       \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 0.02:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 0.02)
     (/ (* i (+ alpha i)) (pow beta 2.0))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.02) {
		tmp = (i * (alpha + i)) / pow(beta, 2.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((alpha + beta) + i)
    if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0d0)) <= 0.02d0) then
        tmp = (i * (alpha + i)) / (beta ** 2.0d0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.02) {
		tmp = (i * (alpha + i)) / Math.pow(beta, 2.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	tmp = 0
	if (((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.02:
		tmp = (i * (alpha + i)) / math.pow(beta, 2.0)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 0.02)
		tmp = Float64(Float64(i * Float64(alpha + i)) / (beta ^ 2.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	tmp = 0.0;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.02)
		tmp = (i * (alpha + i)) / (beta ^ 2.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 14.6

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

   10. Applied rewrites 14.6%

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

   if 0.0200000000000000004 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   3. Applied rewrites 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. div-flip N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. add-to-fraction N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.5%

       \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.5% accurate, 75.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

NOTE: alpha, beta, and i 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr-1 N/A

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

3. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. div-flip N/A

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

   13. lift-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. add-to-fraction N/A

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

   18. lift-/.f64 N/A

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

   19. lift-+.f64 N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
9. Step-by-step derivation

10. Applied rewrites 71.5%

    \[\leadsto \color{blue}{0.0625} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))