Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 99.6%

Time: 6.3s

Alternatives: 13

Speedup: 75.4×

Specification

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

Math

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

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: 16.5% accurate, 1.0× speedup

Math

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

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: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. count-2-rev N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7%

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+l+ N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.7%

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

7. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. count-2-rev N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7%

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+l+ N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.7%

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

9. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.7%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.7%

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

11. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.7%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.6%

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

13. Applied rewrites 99.6%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.6%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.6%

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

15. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6%

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

5. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6%

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (i + fmax(alpha, beta)) + i;
	double t_1 = i / t_0;
	double t_2 = (fmax(alpha, beta) + fmin(alpha, beta)) + i;
	return (fma(t_1, t_2, (fmax(alpha, beta) * (fmin(alpha, beta) / t_0))) / (t_0 - 1.0)) * ((t_2 * t_1) / (t_0 - -1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6%

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. count-2-rev N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7%

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+l+ N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.7%

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

7. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. count-2-rev N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-+.f64 99.7%

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+l+ N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.7%

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

9. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.7%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.7%

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

11. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.7%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.6%

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

13. Applied rewrites 99.6%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. count-2-rev N/A

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

    7. associate-+r+ N/A

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

    8. lower-+.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-+.f64 N/A

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

    12. lower-+.f64 99.6%

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

    13. lift-+.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. associate-+l+ N/A

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

    16. +-commutative N/A

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

    17. lower-+.f64 N/A

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

    18. lower-+.f64 99.6%

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

15. Applied rewrites 99.6%

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

16. Taylor expanded in alpha around 0

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

18. Applied rewrites 86.2%

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

19. Taylor expanded in alpha around 0

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

21. Applied rewrites 85.2%

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

22. Taylor expanded in alpha around 0

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

24. Applied rewrites 84.9%

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

25. Taylor expanded in alpha around 0

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

27. Applied rewrites 84.7%

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

28. Taylor expanded in alpha around 0

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

30. Applied rewrites 84.6%

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

Alternative 5: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

4. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 40.3%

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

6. Applied rewrites 40.3%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 36.5%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 40.2%

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

13. Taylor expanded in alpha around 0

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

15. Applied rewrites 36.4%

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

16. Taylor expanded in alpha around 0

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

18. Applied rewrites 36.2%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l/ N/A

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

    4. lift-+.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. +-commutative N/A

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

    7. lift-fma.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. lower-*.f64 83.9%

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

20. Applied rewrites 83.9%

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

Alternative 6: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

3. Applied rewrites 42.6%

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

4. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 40.3%

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

6. Applied rewrites 40.3%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 36.5%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 40.2%

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

13. Taylor expanded in alpha around 0

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

15. Applied rewrites 36.4%

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

16. Taylor expanded in alpha around 0

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

18. Applied rewrites 36.2%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 83.9%

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

    6. lift-+.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. +-commutative N/A

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

    9. lift-fma.f64 83.9%

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

20. Applied rewrites 83.9%

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

Alternative 7: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.1 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{min}\left(\alpha, \beta\right), -1 \cdot i\right)}{t\_0 - 1} \cdot \frac{\left(\mathsf{max}\left(\alpha, \beta\right) + i\right) \cdot \frac{i}{t\_0}}{t\_0 - -1}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (fmax alpha beta))))
   (if (<= (fmax alpha beta) 1.1e+135)
     0.0625
     (*
      (/ (* -1.0 (fma -1.0 (fmin alpha beta) (* -1.0 i))) (- t_0 1.0))
      (/ (* (+ (fmax alpha beta) i) (/ i t_0)) (- t_0 -1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, fmax(alpha, beta));
	double tmp;
	if (fmax(alpha, beta) <= 1.1e+135) {
		tmp = 0.0625;
	} else {
		tmp = ((-1.0 * fma(-1.0, fmin(alpha, beta), (-1.0 * i))) / (t_0 - 1.0)) * (((fmax(alpha, beta) + i) * (i / t_0)) / (t_0 - -1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.1e+135)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(-1.0 * fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i))) / Float64(t_0 - 1.0)) * Float64(Float64(Float64(fmax(alpha, beta) + i) * Float64(i / t_0)) / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.1e+135], 0.0625, N[(N[(N[(-1.0 * N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1e135

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.6%

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

   if 1.1e135 < beta

   1. Initial program 16.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 42.6%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 40.3%

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

   6. Applied rewrites 40.3%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 36.5%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 40.2%

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

   13. Taylor expanded in alpha around 0

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

   15. Applied rewrites 36.4%

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

   16. Taylor expanded in alpha around 0

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

   18. Applied rewrites 36.2%

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

   19. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 27.9%

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

   21. Applied rewrites 27.9%

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

Alternative 8: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{\mathsf{fma}\left(-1, \mathsf{min}\left(\alpha, \beta\right), -1 \cdot i\right)}{\mathsf{max}\left(\alpha, \beta\right)}\right) \cdot \frac{\left(\mathsf{max}\left(\alpha, \beta\right) + i\right) \cdot \frac{i}{t\_0}}{t\_0 - -1}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (fmax alpha beta))))
   (if (<= (fmax alpha beta) 5.5e+143)
     0.0625
     (*
      (* -1.0 (/ (fma -1.0 (fmin alpha beta) (* -1.0 i)) (fmax alpha beta)))
      (/ (* (+ (fmax alpha beta) i) (/ i t_0)) (- t_0 -1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, fmax(alpha, beta));
	double tmp;
	if (fmax(alpha, beta) <= 5.5e+143) {
		tmp = 0.0625;
	} else {
		tmp = (-1.0 * (fma(-1.0, fmin(alpha, beta), (-1.0 * i)) / fmax(alpha, beta))) * (((fmax(alpha, beta) + i) * (i / t_0)) / (t_0 - -1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 5.5e+143)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(-1.0 * Float64(fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i)) / fmax(alpha, beta))) * Float64(Float64(Float64(fmax(alpha, beta) + i) * Float64(i / t_0)) / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5.5e+143], 0.0625, N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4999999999999997e143

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.6%

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

   if 5.4999999999999997e143 < beta

   1. Initial program 16.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 42.6%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 40.3%

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

   6. Applied rewrites 40.3%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 36.5%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 40.2%

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

   13. Taylor expanded in alpha around 0

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

   15. Applied rewrites 36.4%

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

   16. Taylor expanded in alpha around 0

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

   18. Applied rewrites 36.2%

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

   19. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-*.f64 16.7%

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

   21. Applied rewrites 16.7%

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

Alternative 9: 85.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{min}\left(\alpha, \beta\right), -1 \cdot i\right)}{\mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - 1} \cdot \frac{i}{1 + \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 5.5e+143)
   0.0625
   (*
    (/
     (* -1.0 (fma -1.0 (fmin alpha beta) (* -1.0 i)))
     (- (fma 2.0 i (+ (fmax alpha beta) (fmin alpha beta))) 1.0))
    (/ i (+ 1.0 (+ (fmin alpha beta) (fmax alpha beta)))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 5.5e+143) {
		tmp = 0.0625;
	} else {
		tmp = ((-1.0 * fma(-1.0, fmin(alpha, beta), (-1.0 * i))) / (fma(2.0, i, (fmax(alpha, beta) + fmin(alpha, beta))) - 1.0)) * (i / (1.0 + (fmin(alpha, beta) + fmax(alpha, beta))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5.5e+143)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(-1.0 * fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i))) / Float64(fma(2.0, i, Float64(fmax(alpha, beta) + fmin(alpha, beta))) - 1.0)) * Float64(i / Float64(1.0 + Float64(fmin(alpha, beta) + fmax(alpha, beta)))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5.5e+143], 0.0625, N[(N[(N[(-1.0 * N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(1.0 + N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4999999999999997e143

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.6%

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

   if 5.4999999999999997e143 < beta

   1. Initial program 16.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 42.6%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 16.0%

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 19.2%

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

   9. Applied rewrites 19.2%

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

Alternative 10: 85.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{\mathsf{fma}\left(-1, \mathsf{min}\left(\alpha, \beta\right), -1 \cdot i\right)}{\mathsf{max}\left(\alpha, \beta\right)}\right) \cdot \frac{i}{1 + \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 5.5e+143)
   0.0625
   (*
    (* -1.0 (/ (fma -1.0 (fmin alpha beta) (* -1.0 i)) (fmax alpha beta)))
    (/ i (+ 1.0 (+ (fmin alpha beta) (fmax alpha beta)))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 5.5e+143) {
		tmp = 0.0625;
	} else {
		tmp = (-1.0 * (fma(-1.0, fmin(alpha, beta), (-1.0 * i)) / fmax(alpha, beta))) * (i / (1.0 + (fmin(alpha, beta) + fmax(alpha, beta))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5.5e+143)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(-1.0 * Float64(fma(-1.0, fmin(alpha, beta), Float64(-1.0 * i)) / fmax(alpha, beta))) * Float64(i / Float64(1.0 + Float64(fmin(alpha, beta) + fmax(alpha, beta)))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5.5e+143], 0.0625, N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision] + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(1.0 + N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4999999999999997e143

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.6%

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

   if 5.4999999999999997e143 < beta

   1. Initial program 16.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 42.6%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 16.0%

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 16.4%

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

   9. Applied rewrites 16.4%

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

Alternative 11: 77.2% accurate, 2.4× speedup

Math

\[\left(0.0625 + 0.125 \cdot \frac{\mathsf{max}\left(\alpha, \beta\right)}{i}\right) - 0.125 \cdot \frac{\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)}{i} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.125 (/ (fmax alpha beta) i)))
  (* 0.125 (/ (+ (fmin alpha beta) (fmax alpha beta)) i))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.125 * (fmax(alpha, beta) / i))) - (0.125 * ((fmin(alpha, beta) + fmax(alpha, beta)) / i));
}

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 + (0.125d0 * (fmax(alpha, beta) / i))) - (0.125d0 * ((fmin(alpha, beta) + fmax(alpha, beta)) / i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.125 * (fmax(alpha, beta) / i))) - (0.125 * ((fmin(alpha, beta) + fmax(alpha, beta)) / i));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.125 * (fmax(alpha, beta) / i))) - (0.125 * ((fmin(alpha, beta) + fmax(alpha, beta)) / i))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.125 * Float64(fmax(alpha, beta) / i))) - Float64(0.125 * Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) / i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.125 * (max(alpha, beta) / i))) - (0.125 * ((min(alpha, beta) + max(alpha, beta)) / i));
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.125 * N[(N[Max[alpha, beta], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

   9. lower-+.f64 77.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

4. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \left(0.0625 + \frac{1}{8} \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   2. lower-/.f64 72.9%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied rewrites 72.9%

   \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
8. Add Preprocessing

Alternative 12: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.1 \cdot 10^{+249}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)}{i}, -0.125, 0.125 \cdot \frac{\mathsf{max}\left(\alpha, \beta\right)}{i}\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 1.1e+249)
   0.0625
   (fma
    (/ (+ (fmin alpha beta) (fmax alpha beta)) i)
    -0.125
    (* 0.125 (/ (fmax alpha beta) i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 1.1e+249) {
		tmp = 0.0625;
	} else {
		tmp = fma(((fmin(alpha, beta) + fmax(alpha, beta)) / i), -0.125, (0.125 * (fmax(alpha, beta) / i)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.1e+249)
		tmp = 0.0625;
	else
		tmp = fma(Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) / i), -0.125, Float64(0.125 * Float64(fmax(alpha, beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.1e+249], 0.0625, N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * -0.125 + N[(0.125 * N[(N[Max[alpha, beta], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0999999999999999e249

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.6%

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

   if 1.0999999999999999e249 < beta

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around inf

      \[\leadsto \frac{1}{8} \cdot \frac{\alpha}{i} - \color{blue}{0.125} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{\alpha}{i} - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 6.5%

         \[\leadsto 0.125 \cdot \frac{\alpha}{i} - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 6.5%

      \[\leadsto 0.125 \cdot \frac{\alpha}{i} - \color{blue}{0.125} \cdot \frac{\alpha + \beta}{i} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{\alpha}{i} - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{\alpha}{i} - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

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

         \[\leadsto \frac{1}{8} \cdot \frac{\alpha}{i} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} + \color{blue}{\frac{1}{8} \cdot \frac{\alpha}{i}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\alpha + \beta}{i} \cdot \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) + \color{blue}{\frac{1}{8}} \cdot \frac{\alpha}{i} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, \color{blue}{\mathsf{neg}\left(\frac{1}{8}\right)}, \frac{1}{8} \cdot \frac{\alpha}{i}\right) \]

      7. metadata-eval 6.5%

         \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, 0.125 \cdot \frac{\alpha}{i}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\alpha}{i}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, \frac{-1}{8}, \frac{\alpha}{i} \cdot \frac{1}{8}\right) \]

      10. lower-*.f64 6.5%

          \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, \frac{\alpha}{i} \cdot 0.125\right) \]

   9. Applied rewrites 6.5%

      \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, \color{blue}{-0.125}, \frac{\alpha}{i} \cdot 0.125\right) \]

   10. Taylor expanded in beta around inf

       \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, \frac{1}{8} \cdot \frac{\beta}{i}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       2. lower-/.f64 6.1%

          \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, 0.125 \cdot \frac{\beta}{i}\right) \]

   12. Applied rewrites 6.1%

       \[\leadsto \mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, 0.125 \cdot \frac{\beta}{i}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.6% accurate, 75.4× speedup

Math

\[0.0625 \]

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 16.5%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.6%

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

Reproduce

herbie shell --seed 2025183 
(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)))