Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 84.9%

Time: 12.0s

Alternatives: 13

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

real(8) function code(alpha, beta, i)
    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: 16.1% 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

real(8) function code(alpha, beta, i)
    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: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \frac{\alpha + \beta}{i}\\ t_1 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_2 := \left(\alpha + \beta\right) + 1\\ t_3 := \mathsf{fma}\left(-1, \left(t\_2 + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25, t\_0, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right) \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_0 \cdot t\_2 + t\_3, \frac{t\_3}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_1 - 1}{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(2, i, \alpha\right)}{\beta} \cdot \left(\alpha + i\right)}\right)}^{-1} \cdot {\left(\frac{t\_1 + 1}{\frac{i}{t\_1} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ alpha beta) i))
        (t_1 (fma 2.0 i (+ alpha beta)))
        (t_2 (+ (+ alpha beta) 1.0))
        (t_3
         (fma -1.0 (* (+ t_2 (+ alpha beta)) -2.0) (* -4.0 (+ alpha beta)))))
   (if (<= beta 1.85e+152)
     (*
      (-
       (fma 0.25 t_0 0.25)
       (* (/ (* (+ (- (+ alpha beta) 1.0) (+ alpha beta)) 2.0) i) 0.0625))
      (pow
       (fma
        -1.0
        (/ (fma -1.0 (+ (* t_0 t_2) t_3) (* (/ t_3 i) (+ alpha beta))) i)
        4.0)
       -1.0))
     (*
      (pow
       (/
        (- t_1 1.0)
        (-
         (+ (fma i (/ (+ alpha i) beta) i) alpha)
         (* (/ (fma 2.0 i alpha) beta) (+ alpha i))))
       -1.0)
      (pow (/ (+ t_1 1.0) (* (/ i t_1) (+ (+ alpha beta) i))) -1.0)))))

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 t_2 = (alpha + beta) + 1.0;
	double t_3 = fma(-1.0, ((t_2 + (alpha + beta)) * -2.0), (-4.0 * (alpha + beta)));
	double tmp;
	if (beta <= 1.85e+152) {
		tmp = (fma(0.25, t_0, 0.25) - ((((((alpha + beta) - 1.0) + (alpha + beta)) * 2.0) / i) * 0.0625)) * pow(fma(-1.0, (fma(-1.0, ((t_0 * t_2) + t_3), ((t_3 / i) * (alpha + beta))) / i), 4.0), -1.0);
	} else {
		tmp = pow(((t_1 - 1.0) / ((fma(i, ((alpha + i) / beta), i) + alpha) - ((fma(2.0, i, alpha) / beta) * (alpha + i)))), -1.0) * pow(((t_1 + 1.0) / ((i / t_1) * ((alpha + beta) + i))), -1.0);
	}
	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))
	t_2 = Float64(Float64(alpha + beta) + 1.0)
	t_3 = fma(-1.0, Float64(Float64(t_2 + Float64(alpha + beta)) * -2.0), Float64(-4.0 * Float64(alpha + beta)))
	tmp = 0.0
	if (beta <= 1.85e+152)
		tmp = Float64(Float64(fma(0.25, t_0, 0.25) - Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) - 1.0) + Float64(alpha + beta)) * 2.0) / i) * 0.0625)) * (fma(-1.0, Float64(fma(-1.0, Float64(Float64(t_0 * t_2) + t_3), Float64(Float64(t_3 / i) * Float64(alpha + beta))) / i), 4.0) ^ -1.0));
	else
		tmp = Float64((Float64(Float64(t_1 - 1.0) / Float64(Float64(fma(i, Float64(Float64(alpha + i) / beta), i) + alpha) - Float64(Float64(fma(2.0, i, alpha) / beta) * Float64(alpha + i)))) ^ -1.0) * (Float64(Float64(t_1 + 1.0) / Float64(Float64(i / t_1) * Float64(Float64(alpha + beta) + i))) ^ -1.0));
	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]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[(N[(t$95$2 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(-4.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.85e+152], N[(N[(N[(0.25 * t$95$0 + 0.25), $MachinePrecision] - N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / i), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[(-1.0 * N[(N[(-1.0 * N[(N[(t$95$0 * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[(t$95$3 / i), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 4.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(t$95$1 - 1.0), $MachinePrecision] / N[(N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] + i), $MachinePrecision] + alpha), $MachinePrecision] - N[(N[(N[(2.0 * i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(t$95$1 + 1.0), $MachinePrecision] / N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.84999999999999998e152

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

      \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\alpha + \beta}{i} + \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{\alpha + \beta}{i}, \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\alpha + \beta}{i}}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      12. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      14. lower--.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      15. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

      16. lower-+.f64 85.0

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right)} \]

   11. Taylor expanded in i around -inf

       \[\leadsto {\color{blue}{\left(4 + -1 \cdot \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i}\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto {\color{blue}{\left(-1 \cdot \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i} + 4\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-1, \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i}, 4\right)\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

   13. Applied rewrites 85.1%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right) + \left(\left(\beta + \alpha\right) + 1\right) \cdot \frac{\beta + \alpha}{i}, 1 \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right)}{i}\right)\right)}{i}, 4\right)\right)}}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

   if 1.84999999999999998e152 < 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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 60.0

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

   10. Applied rewrites 60.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right) \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\alpha + \beta}{i} \cdot \left(\left(\alpha + \beta\right) + 1\right) + \mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right), \frac{\mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(2, i, \alpha\right)}{\beta} \cdot \left(\alpha + i\right)}\right)}^{-1} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := \frac{\alpha + \beta}{i}\\ t_2 := \left(\alpha + \beta\right) + 1\\ t_3 := \mathsf{fma}\left(-1, \left(t\_2 + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25, t\_1, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right) \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_1 \cdot t\_2 + t\_3, \frac{t\_3}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot {\left(\frac{t\_0 + 1}{\frac{i}{t\_0} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta)))
        (t_1 (/ (+ alpha beta) i))
        (t_2 (+ (+ alpha beta) 1.0))
        (t_3
         (fma -1.0 (* (+ t_2 (+ alpha beta)) -2.0) (* -4.0 (+ alpha beta)))))
   (if (<= beta 1.85e+152)
     (*
      (-
       (fma 0.25 t_1 0.25)
       (* (/ (* (+ (- (+ alpha beta) 1.0) (+ alpha beta)) 2.0) i) 0.0625))
      (pow
       (fma
        -1.0
        (/ (fma -1.0 (+ (* t_1 t_2) t_3) (* (/ t_3 i) (+ alpha beta))) i)
        4.0)
       -1.0))
     (*
      (/
       (-
        (+ (fma i (/ (+ alpha i) beta) i) alpha)
        (* (/ (- (fma 4.0 i (* alpha 2.0)) 1.0) beta) (+ alpha i)))
       beta)
      (pow (/ (+ t_0 1.0) (* (/ i t_0) (+ (+ alpha beta) i))) -1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = (alpha + beta) / i;
	double t_2 = (alpha + beta) + 1.0;
	double t_3 = fma(-1.0, ((t_2 + (alpha + beta)) * -2.0), (-4.0 * (alpha + beta)));
	double tmp;
	if (beta <= 1.85e+152) {
		tmp = (fma(0.25, t_1, 0.25) - ((((((alpha + beta) - 1.0) + (alpha + beta)) * 2.0) / i) * 0.0625)) * pow(fma(-1.0, (fma(-1.0, ((t_1 * t_2) + t_3), ((t_3 / i) * (alpha + beta))) / i), 4.0), -1.0);
	} else {
		tmp = (((fma(i, ((alpha + i) / beta), i) + alpha) - (((fma(4.0, i, (alpha * 2.0)) - 1.0) / beta) * (alpha + i))) / beta) * pow(((t_0 + 1.0) / ((i / t_0) * ((alpha + beta) + i))), -1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) / i)
	t_2 = Float64(Float64(alpha + beta) + 1.0)
	t_3 = fma(-1.0, Float64(Float64(t_2 + Float64(alpha + beta)) * -2.0), Float64(-4.0 * Float64(alpha + beta)))
	tmp = 0.0
	if (beta <= 1.85e+152)
		tmp = Float64(Float64(fma(0.25, t_1, 0.25) - Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) - 1.0) + Float64(alpha + beta)) * 2.0) / i) * 0.0625)) * (fma(-1.0, Float64(fma(-1.0, Float64(Float64(t_1 * t_2) + t_3), Float64(Float64(t_3 / i) * Float64(alpha + beta))) / i), 4.0) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(fma(i, Float64(Float64(alpha + i) / beta), i) + alpha) - Float64(Float64(Float64(fma(4.0, i, Float64(alpha * 2.0)) - 1.0) / beta) * Float64(alpha + i))) / beta) * (Float64(Float64(t_0 + 1.0) / Float64(Float64(i / t_0) * Float64(Float64(alpha + beta) + i))) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[(N[(t$95$2 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(-4.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.85e+152], N[(N[(N[(0.25 * t$95$1 + 0.25), $MachinePrecision] - N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / i), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[(-1.0 * N[(N[(-1.0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[(t$95$3 / i), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 4.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] + i), $MachinePrecision] + alpha), $MachinePrecision] - N[(N[(N[(N[(4.0 * i + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[Power[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.84999999999999998e152

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

      \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\alpha + \beta}{i} + \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{\alpha + \beta}{i}, \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\alpha + \beta}{i}}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      12. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      14. lower--.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      15. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

      16. lower-+.f64 85.0

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right)} \]

   11. Taylor expanded in i around -inf

       \[\leadsto {\color{blue}{\left(4 + -1 \cdot \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i}\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto {\color{blue}{\left(-1 \cdot \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i} + 4\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-1, \frac{\left(-1 \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right) + -1 \cdot \frac{\left(1 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}{i}\right) - -1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-1 \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right) - 4 \cdot \left(\alpha + \beta\right)\right)}{i}}{i}, 4\right)\right)}}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

   13. Applied rewrites 85.1%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right) + \left(\left(\beta + \alpha\right) + 1\right) \cdot \frac{\beta + \alpha}{i}, 1 \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right)}{i}\right)\right)}{i}, 4\right)\right)}}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right) \]

   if 1.84999999999999998e152 < 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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 59.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right) \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\alpha + \beta}{i} \cdot \left(\left(\alpha + \beta\right) + 1\right) + \mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right), \frac{\mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 1\\ t_1 := \mathsf{fma}\left(-1, \left(t\_0 + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)\\ t_2 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;0.25 \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\alpha + \beta}{i} \cdot t\_0 + t\_1, \frac{t\_1}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot {\left(\frac{t\_2 + 1}{\frac{i}{t\_2} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 1.0))
        (t_1
         (fma -1.0 (* (+ t_0 (+ alpha beta)) -2.0) (* -4.0 (+ alpha beta))))
        (t_2 (fma 2.0 i (+ alpha beta))))
   (if (<= beta 1.95e+152)
     (*
      0.25
      (pow
       (fma
        -1.0
        (/
         (fma
          -1.0
          (+ (* (/ (+ alpha beta) i) t_0) t_1)
          (* (/ t_1 i) (+ alpha beta)))
         i)
        4.0)
       -1.0))
     (*
      (/
       (-
        (+ (fma i (/ (+ alpha i) beta) i) alpha)
        (* (/ (- (fma 4.0 i (* alpha 2.0)) 1.0) beta) (+ alpha i)))
       beta)
      (pow (/ (+ t_2 1.0) (* (/ i t_2) (+ (+ alpha beta) i))) -1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + 1.0;
	double t_1 = fma(-1.0, ((t_0 + (alpha + beta)) * -2.0), (-4.0 * (alpha + beta)));
	double t_2 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 1.95e+152) {
		tmp = 0.25 * pow(fma(-1.0, (fma(-1.0, ((((alpha + beta) / i) * t_0) + t_1), ((t_1 / i) * (alpha + beta))) / i), 4.0), -1.0);
	} else {
		tmp = (((fma(i, ((alpha + i) / beta), i) + alpha) - (((fma(4.0, i, (alpha * 2.0)) - 1.0) / beta) * (alpha + i))) / beta) * pow(((t_2 + 1.0) / ((i / t_2) * ((alpha + beta) + i))), -1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + 1.0)
	t_1 = fma(-1.0, Float64(Float64(t_0 + Float64(alpha + beta)) * -2.0), Float64(-4.0 * Float64(alpha + beta)))
	t_2 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.95e+152)
		tmp = Float64(0.25 * (fma(-1.0, Float64(fma(-1.0, Float64(Float64(Float64(Float64(alpha + beta) / i) * t_0) + t_1), Float64(Float64(t_1 / i) * Float64(alpha + beta))) / i), 4.0) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(fma(i, Float64(Float64(alpha + i) / beta), i) + alpha) - Float64(Float64(Float64(fma(4.0, i, Float64(alpha * 2.0)) - 1.0) / beta) * Float64(alpha + i))) / beta) * (Float64(Float64(t_2 + 1.0) / Float64(Float64(i / t_2) * Float64(Float64(alpha + beta) + i))) ^ -1.0));
	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] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(N[(t$95$0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(-4.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.95e+152], N[(0.25 * N[Power[N[(-1.0 * N[(N[(-1.0 * N[(N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(t$95$1 / i), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 4.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] + i), $MachinePrecision] + alpha), $MachinePrecision] - N[(N[(N[(N[(4.0 * i + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[Power[N[(N[(t$95$2 + 1.0), $MachinePrecision] / N[(N[(i / t$95$2), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.95000000000000006e152

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 79.8%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{0.25} \]

   11. Taylor expanded in i around -inf

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

   13. Applied rewrites 85.8%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right) + \left(\left(\beta + \alpha\right) + 1\right) \cdot \frac{\beta + \alpha}{i}, 1 \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\mathsf{fma}\left(-1, -2 \cdot \left(\left(\left(\beta + \alpha\right) + 1\right) + \left(\beta + \alpha\right)\right), -4 \cdot \left(\beta + \alpha\right)\right)}{i}\right)\right)}{i}, 4\right)\right)}}^{-1} \cdot 0.25 \]

   if 1.95000000000000006e152 < 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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 59.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;0.25 \cdot {\left(\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\alpha + \beta}{i} \cdot \left(\left(\alpha + \beta\right) + 1\right) + \mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right), \frac{\mathsf{fma}\left(-1, \left(\left(\left(\alpha + \beta\right) + 1\right) + \left(\alpha + \beta\right)\right) \cdot -2, -4 \cdot \left(\alpha + \beta\right)\right)}{i} \cdot \left(\alpha + \beta\right)\right)}{i}, 4\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := {\left(\frac{t\_0 + 1}{\frac{i}{t\_0} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot t\_1\\ \end{array} \end{array} \]

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 (fma 2.0 i (+ alpha beta)))
        (t_1 (pow (/ (+ t_0 1.0) (* (/ i t_0) (+ (+ alpha beta) i))) -1.0)))
   (if (<= beta 1.85e+152)
     (*
      t_1
      (-
       (fma 0.25 (/ (+ alpha beta) i) 0.25)
       (* (/ (* (+ (- (+ alpha beta) 1.0) (+ alpha beta)) 2.0) i) 0.0625)))
     (*
      (/
       (-
        (+ (fma i (/ (+ alpha i) beta) i) alpha)
        (* (/ (- (fma 4.0 i (* alpha 2.0)) 1.0) beta) (+ alpha i)))
       beta)
      t_1))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = pow(((t_0 + 1.0) / ((i / t_0) * ((alpha + beta) + i))), -1.0);
	double tmp;
	if (beta <= 1.85e+152) {
		tmp = t_1 * (fma(0.25, ((alpha + beta) / i), 0.25) - ((((((alpha + beta) - 1.0) + (alpha + beta)) * 2.0) / i) * 0.0625));
	} else {
		tmp = (((fma(i, ((alpha + i) / beta), i) + alpha) - (((fma(4.0, i, (alpha * 2.0)) - 1.0) / beta) * (alpha + i))) / beta) * t_1;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(Float64(t_0 + 1.0) / Float64(Float64(i / t_0) * Float64(Float64(alpha + beta) + i))) ^ -1.0
	tmp = 0.0
	if (beta <= 1.85e+152)
		tmp = Float64(t_1 * Float64(fma(0.25, Float64(Float64(alpha + beta) / i), 0.25) - Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) - 1.0) + Float64(alpha + beta)) * 2.0) / i) * 0.0625)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(i, Float64(Float64(alpha + i) / beta), i) + alpha) - Float64(Float64(Float64(fma(4.0, i, Float64(alpha * 2.0)) - 1.0) / beta) * Float64(alpha + i))) / beta) * 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[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[beta, 1.85e+152], N[(t$95$1 * N[(N[(0.25 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] + 0.25), $MachinePrecision] - N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / i), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] + i), $MachinePrecision] + alpha), $MachinePrecision] - N[(N[(N[(N[(4.0 * i + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.84999999999999998e152

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

      \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\alpha + \beta}{i} + \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{\alpha + \beta}{i}, \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\alpha + \beta}{i}}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      12. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      14. lower--.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      15. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

      16. lower-+.f64 85.0

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right)} \]

   if 1.84999999999999998e152 < 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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 59.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, \frac{\alpha + i}{\beta}, i\right) + \alpha\right) - \frac{\mathsf{fma}\left(4, i, \alpha \cdot 2\right) - 1}{\beta} \cdot \left(\alpha + i\right)}{\beta} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := {\left(\frac{t\_0 + 1}{\frac{i}{t\_0} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+151}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot t\_1\\ \end{array} \end{array} \]

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 (fma 2.0 i (+ alpha beta)))
        (t_1 (pow (/ (+ t_0 1.0) (* (/ i t_0) (+ (+ alpha beta) i))) -1.0)))
   (if (<= beta 8.8e+151)
     (*
      t_1
      (-
       (fma 0.25 (/ (+ alpha beta) i) 0.25)
       (* (/ (* (+ (- (+ alpha beta) 1.0) (+ alpha beta)) 2.0) i) 0.0625)))
     (* (/ (+ alpha i) beta) t_1))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = pow(((t_0 + 1.0) / ((i / t_0) * ((alpha + beta) + i))), -1.0);
	double tmp;
	if (beta <= 8.8e+151) {
		tmp = t_1 * (fma(0.25, ((alpha + beta) / i), 0.25) - ((((((alpha + beta) - 1.0) + (alpha + beta)) * 2.0) / i) * 0.0625));
	} else {
		tmp = ((alpha + i) / beta) * t_1;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(Float64(t_0 + 1.0) / Float64(Float64(i / t_0) * Float64(Float64(alpha + beta) + i))) ^ -1.0
	tmp = 0.0
	if (beta <= 8.8e+151)
		tmp = Float64(t_1 * Float64(fma(0.25, Float64(Float64(alpha + beta) / i), 0.25) - Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) - 1.0) + Float64(alpha + beta)) * 2.0) / i) * 0.0625)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * 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[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[beta, 8.8e+151], N[(t$95$1 * N[(N[(0.25 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] + 0.25), $MachinePrecision] - N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / i), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.80000000000000027e151

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

      \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\alpha + \beta}{i} + \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{\alpha + \beta}{i}, \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\alpha + \beta}{i}}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      12. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      14. lower--.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      15. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

      16. lower-+.f64 85.0

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right)} \]

   if 8.80000000000000027e151 < 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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.9

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

   10. Applied rewrites 61.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+151}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\alpha + \beta}{i}, 0.25\right) - \frac{\left(\left(\left(\alpha + \beta\right) - 1\right) + \left(\alpha + \beta\right)\right) \cdot 2}{i} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot {\left(\frac{t\_0 + 1}{\frac{i}{t\_0} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= beta 2.15e+143)
     0.0625
     (*
      (/ (+ alpha i) beta)
      (pow (/ (+ t_0 1.0) (* (/ i t_0) (+ (+ alpha beta) i))) -1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 2.15e+143) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * pow(((t_0 + 1.0) / ((i / t_0) * ((alpha + beta) + i))), -1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.15e+143)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * (Float64(Float64(t_0 + 1.0) / Float64(Float64(i / t_0) * Float64(Float64(alpha + beta) + i))) ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.15e+143], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[Power[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15000000000000001e143

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 80.1%

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

   if 2.15000000000000001e143 < beta

   1. Initial program 2.4%

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

   3. Taylor expanded in 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 2.4

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

   5. Applied rewrites 2.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.5

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

   10. Applied rewrites 61.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot {\left(\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 1}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\mathsf{fma}\left(\alpha + \beta, 2, -1\right)}{i}, \mathsf{fma}\left(\frac{\alpha + \beta}{i}, 0.25, 0.25\right)\right) \cdot \frac{\frac{i}{t\_0} \cdot \left(\left(\alpha + \beta\right) + i\right)}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta))))
   (if (<= beta 1.85e+152)
     (*
      (fma
       -0.125
       (/ (fma (+ alpha beta) 2.0 -1.0) i)
       (fma (/ (+ alpha beta) i) 0.25 0.25))
      (/ (* (/ i t_0) (+ (+ alpha beta) i)) (+ t_0 1.0)))
     (* (/ i beta) (/ (+ alpha i) beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (beta <= 1.85e+152) {
		tmp = fma(-0.125, (fma((alpha + beta), 2.0, -1.0) / i), fma(((alpha + beta) / i), 0.25, 0.25)) * (((i / t_0) * ((alpha + beta) + i)) / (t_0 + 1.0));
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.85e+152)
		tmp = Float64(fma(-0.125, Float64(fma(Float64(alpha + beta), 2.0, -1.0) / i), fma(Float64(Float64(alpha + beta) / i), 0.25, 0.25)) * Float64(Float64(Float64(i / t_0) * Float64(Float64(alpha + beta) + i)) / Float64(t_0 + 1.0)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.85e+152], N[(N[(-0.125 * N[(N[(N[(alpha + beta), $MachinePrecision] * 2.0 + -1.0), $MachinePrecision] / i), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.25 + 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.84999999999999998e152

   1. Initial program 19.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 20.0

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

   5. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in i around inf

      \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\alpha + \beta}{i} + \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{\alpha + \beta}{i}, \frac{1}{4}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\alpha + \beta}{i}}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\beta + \alpha}}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      9. distribute-lft-out N/A

         \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      12. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\beta + \alpha\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      14. lower--.f64 N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      15. +-commutative N/A

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\beta + \alpha}{i}, \frac{1}{4}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

      16. lower-+.f64 85.0

          \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\color{blue}{\left(\beta + \alpha\right)} - 1\right)\right)}{i}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto {\left(\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right)}^{-1} \cdot \color{blue}{\left(\mathsf{fma}\left(0.25, \frac{\beta + \alpha}{i}, 0.25\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\beta + \alpha\right) + \left(\left(\beta + \alpha\right) - 1\right)\right)}{i}\right)} \]

   12. Applied rewrites 85.0%

       \[\leadsto \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \mathsf{fma}\left(-0.125, \frac{\mathsf{fma}\left(\alpha + \beta, 2, -1\right)}{i}, \mathsf{fma}\left(\frac{\alpha + \beta}{i}, 0.25, 0.25\right)\right)} \]

   if 1.84999999999999998e152 < 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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\mathsf{fma}\left(\alpha + \beta, 2, -1\right)}{i}, \mathsf{fma}\left(\frac{\alpha + \beta}{i}, 0.25, 0.25\right)\right) \cdot \frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+144}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+144) 0.0625 (* (/ i beta) (/ (+ alpha i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+144) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1d+144) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + 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 <= 1e+144) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1e+144:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+144)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + 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 <= 1e+144)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + 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, 1e+144], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000002e144

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 80.1%

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

   if 1.00000000000000002e144 < beta

   1. Initial program 2.4%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+144}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.85e+152) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.85e+152) {
		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.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.85d+152) 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 <= 1.85e+152) {
		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 <= 1.85e+152:
		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 <= 1.85e+152)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(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 <= 1.85e+152)
		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, 1.85e+152], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.84999999999999998e152

   1. Initial program 19.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.9%

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

   if 1.84999999999999998e152 < 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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 54.3%

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

Alternative 10: 74.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.15e+196) 0.0625 (/ (* (/ i beta) alpha) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.15e+196) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.15d+196) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * alpha) / 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 <= 1.15e+196) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.15e+196:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * alpha) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.15e+196)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * alpha) / 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 <= 1.15e+196)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * alpha) / 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, 1.15e+196], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * alpha), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1499999999999999e196

   1. Initial program 18.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.2%

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

   if 1.1499999999999999e196 < 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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 26.5%

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

   10. Applied rewrites 34.1%

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

Alternative 11: 73.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{\beta}{i} \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.45e+237) 0.0625 (/ alpha (* (/ beta i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.45e+237) {
		tmp = 0.0625;
	} else {
		tmp = alpha / ((beta / i) * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.45d+237) then
        tmp = 0.0625d0
    else
        tmp = alpha / ((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 <= 1.45e+237) {
		tmp = 0.0625;
	} else {
		tmp = alpha / ((beta / i) * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.45e+237:
		tmp = 0.0625
	else:
		tmp = alpha / ((beta / i) * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.45e+237)
		tmp = 0.0625;
	else
		tmp = Float64(alpha / Float64(Float64(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 <= 1.45e+237)
		tmp = 0.0625;
	else
		tmp = alpha / ((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, 1.45e+237], 0.0625, N[(alpha / N[(N[(beta / i), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45000000000000005e237

   1. Initial program 17.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 74.6%

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

   if 1.45000000000000005e237 < 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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 34.5%

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

   10. Applied rewrites 42.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{\beta}{i} \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.45e+237) 0.0625 (* (/ i (* beta beta)) alpha)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.45e+237) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.45d+237) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta * beta)) * alpha
    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 <= 1.45e+237) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.45e+237:
		tmp = 0.0625
	else:
		tmp = (i / (beta * beta)) * alpha
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.45e+237)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta * beta)) * alpha);
	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 <= 1.45e+237)
		tmp = 0.0625;
	else
		tmp = (i / (beta * beta)) * alpha;
	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, 1.45e+237], 0.0625, N[(N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45000000000000005e237

   1. Initial program 17.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 74.6%

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

   if 1.45000000000000005e237 < 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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 34.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.2% 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.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 71.7%

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

Reproduce

herbie shell --seed 2024264 
(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)))