Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 82.8%

Time: 9.3s

Alternatives: 14

Speedup: 115.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \frac{\alpha + \beta}{i}\\ \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+112}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\left(\beta + i\right) \cdot i}{t\_0} \cdot \left(\left(\left(\alpha + i\right) + \beta\right) \cdot \frac{i}{t\_0}\right)}{\mathsf{fma}\left(t\_0, t\_0, -1\right)}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot t\_1\right)\right) - 0.125 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (/ (+ alpha beta) i)))
   (if (<= beta 1.1e+112)
     0.0625
     (if (<= beta 1.4e+141)
       (/
        (* (/ (* (+ beta i) i) t_0) (* (+ (+ alpha i) beta) (/ i t_0)))
        (fma t_0 t_0 -1.0))
       (if (<= beta 5.5e+217)
         (- (- 0.0625 (* -0.0625 (* 2.0 t_1))) (* 0.125 t_1))
         (/ (* (/ (+ alpha i) beta) i) beta))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (alpha + beta) / i;
	double tmp;
	if (beta <= 1.1e+112) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+141) {
		tmp = ((((beta + i) * i) / t_0) * (((alpha + i) + beta) * (i / t_0))) / fma(t_0, t_0, -1.0);
	} else if (beta <= 5.5e+217) {
		tmp = (0.0625 - (-0.0625 * (2.0 * t_1))) - (0.125 * t_1);
	} else {
		tmp = (((alpha + i) / beta) * i) / beta;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(alpha + beta) / i)
	tmp = 0.0
	if (beta <= 1.1e+112)
		tmp = 0.0625;
	elseif (beta <= 1.4e+141)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + i) * i) / t_0) * Float64(Float64(Float64(alpha + i) + beta) * Float64(i / t_0))) / fma(t_0, t_0, -1.0));
	elseif (beta <= 5.5e+217)
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(2.0 * t_1))) - Float64(0.125 * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + i) / beta) * i) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[beta, 1.1e+112], 0.0625, If[LessEqual[beta, 1.4e+141], N[(N[(N[(N[(N[(beta + i), $MachinePrecision] * i), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(alpha + i), $MachinePrecision] + beta), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5.5e+217], N[(N[(0.0625 - N[(-0.0625 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.1e112

   1. Initial program 14.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.4%

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

   if 1.1e112 < beta < 1.39999999999999996e141

   1. Initial program 2.2%

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

   3. Taylor expanded in alpha around 0

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

   5. Applied rewrites 2.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. difference-of-sqr--1-rev N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. metadata-eval 2.2

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

   7. Applied rewrites 2.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 63.4%

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

   if 1.39999999999999996e141 < beta < 5.5e217

   1. Initial program 0.1%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 0.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 0.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 58.1%

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

   if 5.5e217 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+112}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\left(\beta + i\right) \cdot i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\left(\left(\alpha + i\right) + \beta\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), -1\right)}\\ \mathbf{elif}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot \frac{\alpha + \beta}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \frac{\alpha + \beta}{i}\\ t_3 := \left(\alpha + \beta\right) + i\\ t_4 := i \cdot t\_3\\ t_5 := \left(\beta + \alpha\right) + i\\ t_6 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(\beta \cdot \alpha + t\_4\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_5, i, \beta \cdot \alpha\right)}{t\_6} \cdot \left(t\_5 \cdot \frac{i}{t\_6}\right)}{\left(t\_3 + i\right) \cdot t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot t\_2\right)\right) - 0.125 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (/ (+ alpha beta) i))
        (t_3 (+ (+ alpha beta) i))
        (t_4 (* i t_3))
        (t_5 (+ (+ beta alpha) i))
        (t_6 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_4 (+ (* beta alpha) t_4)) t_1) (- t_1 1.0)) INFINITY)
     (/
      (* (/ (fma t_5 i (* beta alpha)) t_6) (* t_5 (/ i t_6)))
      (- (* (+ t_3 i) t_0) 1.0))
     (- (- 0.0625 (* -0.0625 (* 2.0 t_2))) (* 0.125 t_2)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = (alpha + beta) / i;
	double t_3 = (alpha + beta) + i;
	double t_4 = i * t_3;
	double t_5 = (beta + alpha) + i;
	double t_6 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_4 * ((beta * alpha) + t_4)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((fma(t_5, i, (beta * alpha)) / t_6) * (t_5 * (i / t_6))) / (((t_3 + i) * t_0) - 1.0);
	} else {
		tmp = (0.0625 - (-0.0625 * (2.0 * t_2))) - (0.125 * t_2);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(alpha + beta) / i)
	t_3 = Float64(Float64(alpha + beta) + i)
	t_4 = Float64(i * t_3)
	t_5 = Float64(Float64(beta + alpha) + i)
	t_6 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(Float64(beta * alpha) + t_4)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(fma(t_5, i, Float64(beta * alpha)) / t_6) * Float64(t_5 * Float64(i / t_6))) / Float64(Float64(Float64(t_3 + i) * t_0) - 1.0));
	else
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(2.0 * t_2))) - Float64(0.125 * t_2));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(N[(beta * alpha), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(t$95$5 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision] * N[(t$95$5 * N[(i / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$3 + i), $MachinePrecision] * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lower-+.f64 99.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 0.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 73.7%

       \[\leadsto \color{blue}{\left(0.0625 - -0.0625 \cdot \left(2 \cdot \frac{\alpha + \beta}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \frac{\alpha + \beta}{i}\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_4 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_5 := t\_4 \cdot t\_4\\ t_6 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_5}}{t\_5 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, i, \beta \cdot \alpha\right)}{t\_3} \cdot \left(t\_2 \cdot \frac{i}{t\_3}\right)}{\mathsf{fma}\left(t\_6, t\_6, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot t\_1\right)\right) - 0.125 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (/ (+ alpha beta) i))
        (t_2 (+ (+ beta alpha) i))
        (t_3 (fma 2.0 i (+ beta alpha)))
        (t_4 (+ (+ alpha beta) (* 2.0 i)))
        (t_5 (* t_4 t_4))
        (t_6 (fma 2.0 i (+ alpha beta))))
   (if (<= (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_5) (- t_5 1.0)) INFINITY)
     (/
      (* (/ (fma t_2 i (* beta alpha)) t_3) (* t_2 (/ i t_3)))
      (fma t_6 t_6 -1.0))
     (- (- 0.0625 (* -0.0625 (* 2.0 t_1))) (* 0.125 t_1)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) / i;
	double t_2 = (beta + alpha) + i;
	double t_3 = fma(2.0, i, (beta + alpha));
	double t_4 = (alpha + beta) + (2.0 * i);
	double t_5 = t_4 * t_4;
	double t_6 = fma(2.0, i, (alpha + beta));
	double tmp;
	if ((((t_0 * ((beta * alpha) + t_0)) / t_5) / (t_5 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((fma(t_2, i, (beta * alpha)) / t_3) * (t_2 * (i / t_3))) / fma(t_6, t_6, -1.0);
	} else {
		tmp = (0.0625 - (-0.0625 * (2.0 * t_1))) - (0.125 * t_1);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) / i)
	t_2 = Float64(Float64(beta + alpha) + i)
	t_3 = fma(2.0, i, Float64(beta + alpha))
	t_4 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_5 = Float64(t_4 * t_4)
	t_6 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_5) / Float64(t_5 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(fma(t_2, i, Float64(beta * alpha)) / t_3) * Float64(t_2 * Float64(i / t_3))) / fma(t_6, t_6, -1.0));
	else
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(2.0 * t_1))) - Float64(0.125 * t_1));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(t$95$5 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(t$95$2 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$2 * N[(i / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * t$95$6 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 - N[(-0.0625 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. difference-of-sqr--1-rev N/A

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

      5. lower-fma.f64 99.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 99.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.6

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. lift-fma.f64 99.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 0.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 73.7%

       \[\leadsto \color{blue}{\left(0.0625 - -0.0625 \cdot \left(2 \cdot \frac{\alpha + \beta}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{\left(\alpha + i\right) + \beta}{t\_0}}{t\_0 - -1} \cdot \frac{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{i} - -1\right) \cdot i, i, \beta \cdot \alpha\right) \cdot \frac{i}{t\_0}}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= i 3.8e+136)
     (*
      (/ (/ (+ (+ alpha i) beta) t_0) (- t_0 -1.0))
      (/
       (* (fma (* (- (/ (+ beta alpha) i) -1.0) i) i (* beta alpha)) (/ i t_0))
       (- t_0 1.0)))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (i <= 3.8e+136) {
		tmp = ((((alpha + i) + beta) / t_0) / (t_0 - -1.0)) * ((fma(((((beta + alpha) / i) - -1.0) * i), i, (beta * alpha)) * (i / t_0)) / (t_0 - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (i <= 3.8e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + i) + beta) / t_0) / Float64(t_0 - -1.0)) * Float64(Float64(fma(Float64(Float64(Float64(Float64(beta + alpha) / i) - -1.0) * i), i, Float64(beta * alpha)) * Float64(i / t_0)) / Float64(t_0 - 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.8e+136], N[(N[(N[(N[(N[(alpha + i), $MachinePrecision] + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision] - -1.0), $MachinePrecision] * i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if i < 3.80000000000000015e136

   1. Initial program 30.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 30.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 30.4%

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

   9. Applied rewrites 86.1%

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

   if 3.80000000000000015e136 < i

   1. Initial program 0.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{\left(\alpha + i\right) + \beta}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \frac{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{i} - -1\right) \cdot i, i, \beta \cdot \alpha\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i\\ t_1 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1 - -1} \cdot \frac{\mathsf{fma}\left(t\_0, i, \alpha \cdot \beta\right) \cdot \frac{i}{t\_1}}{t\_1 - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) i)) (t_1 (fma 2.0 i (+ alpha beta))))
   (if (<= i 3.8e+136)
     (*
      (/ (/ t_0 t_1) (- t_1 -1.0))
      (/ (* (fma t_0 i (* alpha beta)) (/ i t_1)) (- t_1 1.0)))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + i;
	double t_1 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (i <= 3.8e+136) {
		tmp = ((t_0 / t_1) / (t_1 - -1.0)) * ((fma(t_0, i, (alpha * beta)) * (i / t_1)) / (t_1 - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + i)
	t_1 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (i <= 3.8e+136)
		tmp = Float64(Float64(Float64(t_0 / t_1) / Float64(t_1 - -1.0)) * Float64(Float64(fma(t_0, i, Float64(alpha * beta)) * Float64(i / t_1)) / Float64(t_1 - 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.8e+136], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 * i + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.80000000000000015e136

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.1%

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

   6. Applied rewrites 86.0%

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

   if 3.80000000000000015e136 < i

   1. Initial program 0.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - -1} \cdot \frac{\mathsf{fma}\left(\left(\alpha + \beta\right) + i, i, \alpha \cdot \beta\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i\\ t_1 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{i}{t\_1}}{t\_1 - -1} \cdot \frac{\frac{\mathsf{fma}\left(t\_0, i, \beta \cdot \alpha\right)}{t\_1}}{t\_1 - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) i)) (t_1 (fma 2.0 i (+ beta alpha))))
   (if (<= i 3.8e+136)
     (*
      (/ (* t_0 (/ i t_1)) (- t_1 -1.0))
      (/ (/ (fma t_0 i (* beta alpha)) t_1) (- t_1 1.0)))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + i;
	double t_1 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (i <= 3.8e+136) {
		tmp = ((t_0 * (i / t_1)) / (t_1 - -1.0)) * ((fma(t_0, i, (beta * alpha)) / t_1) / (t_1 - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + i)
	t_1 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (i <= 3.8e+136)
		tmp = Float64(Float64(Float64(t_0 * Float64(i / t_1)) / Float64(t_1 - -1.0)) * Float64(Float64(fma(t_0, i, Float64(beta * alpha)) / t_1) / Float64(t_1 - 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.8e+136], N[(N[(N[(t$95$0 * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.80000000000000015e136

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 86.0%

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

   if 3.80000000000000015e136 < i

   1. Initial program 0.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \frac{\frac{\mathsf{fma}\left(\left(\beta + \alpha\right) + i, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + \beta}{i}\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot t\_0\right)\right) - 0.125 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ alpha beta) i)))
   (if (<= beta 5.5e+217)
     (- (- 0.0625 (* -0.0625 (* 2.0 t_0))) (* 0.125 t_0))
     (/ (* (/ (+ alpha i) beta) i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) / i;
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = (0.0625 - (-0.0625 * (2.0 * t_0))) - (0.125 * t_0);
	} else {
		tmp = (((alpha + i) / beta) * i) / beta;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) / i
    if (beta <= 5.5d+217) then
        tmp = (0.0625d0 - ((-0.0625d0) * (2.0d0 * t_0))) - (0.125d0 * t_0)
    else
        tmp = (((alpha + i) / beta) * i) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) / i;
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = (0.0625 - (-0.0625 * (2.0 * t_0))) - (0.125 * t_0);
	} else {
		tmp = (((alpha + i) / beta) * i) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) / i
	tmp = 0
	if beta <= 5.5e+217:
		tmp = (0.0625 - (-0.0625 * (2.0 * t_0))) - (0.125 * t_0)
	else:
		tmp = (((alpha + i) / beta) * i) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) / i)
	tmp = 0.0
	if (beta <= 5.5e+217)
		tmp = Float64(Float64(0.0625 - Float64(-0.0625 * Float64(2.0 * t_0))) - Float64(0.125 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + i) / beta) * i) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) / i;
	tmp = 0.0;
	if (beta <= 5.5e+217)
		tmp = (0.0625 - (-0.0625 * (2.0 * t_0))) - (0.125 * t_0);
	else
		tmp = (((alpha + i) / beta) * i) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[beta, 5.5e+217], N[(N[(0.0625 - N[(-0.0625 * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e217

   1. Initial program 12.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 12.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 12.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 74.8%

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

   if 5.5e217 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\left(0.0625 - -0.0625 \cdot \left(2 \cdot \frac{\alpha + \beta}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{2}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.5e+217)
   (-
    (fma (* (+ alpha beta) (/ 2.0 i)) 0.0625 0.0625)
    (* 0.125 (/ (+ alpha beta) i)))
   (/ (* (/ (+ alpha i) beta) i) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = fma(((alpha + beta) * (2.0 / i)), 0.0625, 0.0625) - (0.125 * ((alpha + beta) / i));
	} else {
		tmp = (((alpha + i) / beta) * i) / beta;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+217)
		tmp = Float64(fma(Float64(Float64(alpha + beta) * Float64(2.0 / i)), 0.0625, 0.0625) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + i) / beta) * i) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+217], N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(2.0 / i), $MachinePrecision]), $MachinePrecision] * 0.0625 + 0.0625), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e217

   1. Initial program 12.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 12.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 12.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 74.8%

       \[\leadsto \color{blue}{\left(0.0625 - -0.0625 \cdot \left(2 \cdot \frac{\alpha + \beta}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 73.9%

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

   if 5.5e217 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{2}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta} \cdot i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.8e+184) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+184) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.8d+184) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+184) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.8e+184:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.8e+184)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.8e+184)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.8e+184], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999993e184

   1. Initial program 13.1%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.7%

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

   if 4.79999999999999993e184 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 72.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.5e+217) 0.0625 (/ (* (/ i beta) i) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.5d+217) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * i) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.5e+217:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * i) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+217)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * i) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.5e+217)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * i) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+217], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e217

   1. Initial program 12.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 72.4%

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

   if 5.5e217 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 28.4%

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

   13. Applied rewrites 83.0%

       \[\leadsto \frac{\frac{i}{\beta} \cdot i}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.5e+217) 0.0625 (* (/ (/ i beta) beta) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.5d+217) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) / beta) * i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+217) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.5e+217:
		tmp = 0.0625
	else:
		tmp = ((i / beta) / beta) * i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+217)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) / beta) * i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.5e+217)
		tmp = 0.0625;
	else
		tmp = ((i / beta) / beta) * i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+217], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e217

   1. Initial program 12.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 72.4%

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

   if 5.5e217 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 28.4%

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

   13. Applied rewrites 56.0%

       \[\leadsto \frac{\frac{i}{\beta}}{\beta} \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 9.8 \cdot 10^{+39}:\\ \;\;\;\;\left(\alpha + i\right) \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 9.8e+39) (* (+ alpha i) (/ i (* beta beta))) 0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 9.8e+39) {
		tmp = (alpha + i) * (i / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 9.8d+39) then
        tmp = (alpha + i) * (i / (beta * beta))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 9.8e+39) {
		tmp = (alpha + i) * (i / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if i <= 9.8e+39:
		tmp = (alpha + i) * (i / (beta * beta))
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 9.8e+39)
		tmp = Float64(Float64(alpha + i) * Float64(i / Float64(beta * beta)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 9.8e+39)
		tmp = (alpha + i) * (i / (beta * beta));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 9.8e+39], N[(N[(alpha + i), $MachinePrecision] * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 9.79999999999999974e39

   1. Initial program 55.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 43.3%

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

   if 9.79999999999999974e39 < i

   1. Initial program 6.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 72.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9.8 \cdot 10^{+39}:\\ \;\;\;\;\left(\alpha + i\right) \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 9.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 9.4e+39) (/ (* i i) (* beta beta)) 0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 9.4e+39) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 9.4d+39) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 9.4e+39) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if i <= 9.4e+39:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 9.4e+39)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 9.4e+39)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 9.4e+39], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 9.3999999999999998e39

   1. Initial program 55.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 43.4%

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

   if 9.3999999999999998e39 < i

   1. Initial program 6.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 72.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.5% accurate, 115.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 11.5%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 67.1%

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

6. Final simplification 67.1%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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)))