Octave 3.8, jcobi/4

Percentage Accurate: 16.7% → 99.7%

Time: 6.3s

Alternatives: 9

Speedup: 115.0×

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: 16.7% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.7%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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 43.2%

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

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 99.7

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

7. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. associate--l+ N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 81.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.7%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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 43.2%

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

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in beta around inf

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

10. Applied rewrites 81.4%

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

Alternative 3: 81.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.7%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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 43.2%

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

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. associate-*l/ N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/l* N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in beta around inf

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

10. Applied rewrites 81.4%

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

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+113}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_1 - -1} \cdot \frac{\mathsf{fma}\left(\beta, \frac{\alpha}{t\_0}, \frac{i}{t\_0} \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{t\_1 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))) (t_1 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 1.65e+113)
     0.0625
     (*
      (/ i (- t_1 -1.0))
      (/
       (fma beta (/ alpha t_0) (* (/ i t_0) (+ i (+ alpha beta))))
       (- t_1 1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 1.65e+113) {
		tmp = 0.0625;
	} else {
		tmp = (i / (t_1 - -1.0)) * (fma(beta, (alpha / t_0), ((i / t_0) * (i + (alpha + beta)))) / (t_1 - 1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.65e+113)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(t_1 - -1.0)) * Float64(fma(beta, Float64(alpha / t_0), Float64(Float64(i / t_0) * Float64(i + Float64(alpha + beta)))) / Float64(t_1 - 1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.65e+113], 0.0625, N[(N[(i / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * N[(alpha / t$95$0), $MachinePrecision] + N[(N[(i / t$95$0), $MachinePrecision] * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6500000000000002e113

   1. Initial program 20.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 80.8%

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

   if 1.6500000000000002e113 < beta

   1. Initial program 1.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 26.1%

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

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 99.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 99.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 72.9%

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

Alternative 5: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+113}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i\right) + i}}{t\_0 - -1} \cdot \frac{\alpha + i}{t\_0 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 1.65e+113)
     0.0625
     (*
      (/
       (* (+ (+ beta alpha) i) (/ i (+ (+ (+ alpha beta) i) i)))
       (- t_0 -1.0))
      (/ (+ alpha i) (- t_0 1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 1.65e+113) {
		tmp = 0.0625;
	} else {
		tmp = ((((beta + alpha) + i) * (i / (((alpha + beta) + i) + i))) / (t_0 - -1.0)) * ((alpha + i) / (t_0 - 1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.65e+113)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / Float64(Float64(Float64(alpha + beta) + i) + i))) / Float64(t_0 - -1.0)) * Float64(Float64(alpha + i) / Float64(t_0 - 1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.65e+113], 0.0625, N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6500000000000002e113

   1. Initial program 20.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 80.8%

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

   if 1.6500000000000002e113 < beta

   1. Initial program 1.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 26.1%

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

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 99.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 99.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in beta around inf

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

      1. lower-+.f64 65.8

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

   10. Applied rewrites 65.8%

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

Alternative 6: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\left(\left(\alpha + \beta\right) + i\right) + i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+135)
   0.0625
   (*
    (/
     (* (+ (+ beta alpha) i) (/ i (+ (+ (+ alpha beta) i) i)))
     (- (fma 2.0 i (+ beta alpha)) -1.0))
    (/ (+ alpha i) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+135) {
		tmp = 0.0625;
	} else {
		tmp = ((((beta + alpha) + i) * (i / (((alpha + beta) + i) + i))) / (fma(2.0, i, (beta + alpha)) - -1.0)) * ((alpha + i) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+135)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / Float64(Float64(Float64(alpha + beta) + i) + i))) / Float64(fma(2.0, i, Float64(beta + alpha)) - -1.0)) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+135], 0.0625, N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000003e135

   1. Initial program 20.4%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 47.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 80.0%

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

   if 6.5000000000000003e135 < beta

   1. Initial program 0.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 23.9%

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

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 99.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 99.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 66.1

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

   10. Applied rewrites 66.1%

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

Alternative 7: 73.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{i \cdot \left(\alpha + i\right)}{\beta}}{-1 - \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+135)
   0.0625
   (/
    (* -1.0 (/ (* i (+ alpha i)) beta))
    (- -1.0 (fma 2.0 i (+ alpha beta))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+135) {
		tmp = 0.0625;
	} else {
		tmp = (-1.0 * ((i * (alpha + i)) / beta)) / (-1.0 - fma(2.0, i, (alpha + beta)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+135)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(i * Float64(alpha + i)) / beta)) / Float64(-1.0 - fma(2.0, i, Float64(alpha + beta))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+135], 0.0625, N[(N[(-1.0 * N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000003e135

   1. Initial program 20.4%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 47.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 80.0%

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

   if 6.5000000000000003e135 < beta

   1. Initial program 0.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 23.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 23.9%

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

   7. Applied rewrites 9.2%

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

   8. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.7

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

   10. Applied rewrites 45.7%

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

Alternative 8: 72.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+257}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\mathsf{fma}\left(t\_0, t\_0, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 2.6e+257) 0.0625 (/ (* i (+ alpha i)) (fma t_0 t_0 -1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 2.6e+257) {
		tmp = 0.0625;
	} else {
		tmp = (i * (alpha + i)) / fma(t_0, t_0, -1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.6e+257)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(alpha + i)) / fma(t_0, t_0, -1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6e+257], 0.0625, N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000021e257

   1. Initial program 17.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 45.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 27.4%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 74.0%

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

   if 2.60000000000000021e257 < beta

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate1 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. metadata-eval 0.0

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 0.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 0.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 38.4

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

   10. Applied rewrites 38.4%

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

Alternative 9: 70.6% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

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

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
    code = 0.0625d0
end function

Java

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

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.7%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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 43.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

5. Applied rewrites 43.2%

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

7. Applied rewrites 26.1%

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

8. Taylor expanded in i around inf

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

10. Applied rewrites 70.6%

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

Reproduce

herbie shell --seed 2025108 
(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)))