Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 82.7%

Time: 21.1s

Alternatives: 7

Speedup: 53.0×

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: 16.3% 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: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := 0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25\\ \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+77}:\\ \;\;\;\;\left(0.0625 + \left(0.25 \cdot \frac{t_1}{i} - \frac{0.015625 \cdot \left(-1 + {\left(\beta + \alpha\right)}^{2}\right) + \left(\beta + \alpha\right) \cdot \left(0.25 \cdot t_1 - 0.0625 \cdot \left(\beta + \alpha\right)\right)}{{i}^{2}}\right)\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+206}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\beta} \cdot \sqrt{\alpha + i}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* 2.0 i)))
        (t_1
         (- (* 0.25 (+ (* alpha 2.0) (* beta 2.0))) (* (+ beta alpha) 0.25))))
   (if (<= beta 1.32e+77)
     (-
      (+
       0.0625
       (-
        (* 0.25 (/ t_1 i))
        (/
         (+
          (* 0.015625 (+ -1.0 (pow (+ beta alpha) 2.0)))
          (* (+ beta alpha) (- (* 0.25 t_1) (* 0.0625 (+ beta alpha)))))
         (pow i 2.0))))
      (* 0.0625 (/ (+ beta alpha) i)))
     (if (<= beta 1.8e+109)
       (/
        (* i (/ i (/ (pow (+ beta (* 2.0 i)) 2.0) (pow (+ beta i) 2.0))))
        (+ -1.0 (* t_0 t_0)))
       (if (<= beta 3.6e+206)
         (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125))
         (pow (* (/ (sqrt i) beta) (sqrt (+ alpha i))) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = (0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25);
	double tmp;
	if (beta <= 1.32e+77) {
		tmp = (0.0625 + ((0.25 * (t_1 / i)) - (((0.015625 * (-1.0 + pow((beta + alpha), 2.0))) + ((beta + alpha) * ((0.25 * t_1) - (0.0625 * (beta + alpha))))) / pow(i, 2.0)))) - (0.0625 * ((beta + alpha) / i));
	} else if (beta <= 1.8e+109) {
		tmp = (i * (i / (pow((beta + (2.0 * i)), 2.0) / pow((beta + i), 2.0)))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+206) {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	} else {
		tmp = pow(((sqrt(i) / beta) * sqrt((alpha + i))), 2.0);
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + (2.0d0 * i)
    t_1 = (0.25d0 * ((alpha * 2.0d0) + (beta * 2.0d0))) - ((beta + alpha) * 0.25d0)
    if (beta <= 1.32d+77) then
        tmp = (0.0625d0 + ((0.25d0 * (t_1 / i)) - (((0.015625d0 * ((-1.0d0) + ((beta + alpha) ** 2.0d0))) + ((beta + alpha) * ((0.25d0 * t_1) - (0.0625d0 * (beta + alpha))))) / (i ** 2.0d0)))) - (0.0625d0 * ((beta + alpha) / i))
    else if (beta <= 1.8d+109) then
        tmp = (i * (i / (((beta + (2.0d0 * i)) ** 2.0d0) / ((beta + i) ** 2.0d0)))) / ((-1.0d0) + (t_0 * t_0))
    else if (beta <= 3.6d+206) then
        tmp = (0.0625d0 + (0.125d0 * (beta / i))) + ((beta / i) * (-0.125d0))
    else
        tmp = ((sqrt(i) / beta) * sqrt((alpha + i))) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = (0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25);
	double tmp;
	if (beta <= 1.32e+77) {
		tmp = (0.0625 + ((0.25 * (t_1 / i)) - (((0.015625 * (-1.0 + Math.pow((beta + alpha), 2.0))) + ((beta + alpha) * ((0.25 * t_1) - (0.0625 * (beta + alpha))))) / Math.pow(i, 2.0)))) - (0.0625 * ((beta + alpha) / i));
	} else if (beta <= 1.8e+109) {
		tmp = (i * (i / (Math.pow((beta + (2.0 * i)), 2.0) / Math.pow((beta + i), 2.0)))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+206) {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	} else {
		tmp = Math.pow(((Math.sqrt(i) / beta) * Math.sqrt((alpha + i))), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (2.0 * i)
	t_1 = (0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)
	tmp = 0
	if beta <= 1.32e+77:
		tmp = (0.0625 + ((0.25 * (t_1 / i)) - (((0.015625 * (-1.0 + math.pow((beta + alpha), 2.0))) + ((beta + alpha) * ((0.25 * t_1) - (0.0625 * (beta + alpha))))) / math.pow(i, 2.0)))) - (0.0625 * ((beta + alpha) / i))
	elif beta <= 1.8e+109:
		tmp = (i * (i / (math.pow((beta + (2.0 * i)), 2.0) / math.pow((beta + i), 2.0)))) / (-1.0 + (t_0 * t_0))
	elif beta <= 3.6e+206:
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)
	else:
		tmp = math.pow(((math.sqrt(i) / beta) * math.sqrt((alpha + i))), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	t_1 = Float64(Float64(0.25 * Float64(Float64(alpha * 2.0) + Float64(beta * 2.0))) - Float64(Float64(beta + alpha) * 0.25))
	tmp = 0.0
	if (beta <= 1.32e+77)
		tmp = Float64(Float64(0.0625 + Float64(Float64(0.25 * Float64(t_1 / i)) - Float64(Float64(Float64(0.015625 * Float64(-1.0 + (Float64(beta + alpha) ^ 2.0))) + Float64(Float64(beta + alpha) * Float64(Float64(0.25 * t_1) - Float64(0.0625 * Float64(beta + alpha))))) / (i ^ 2.0)))) - Float64(0.0625 * Float64(Float64(beta + alpha) / i)));
	elseif (beta <= 1.8e+109)
		tmp = Float64(Float64(i * Float64(i / Float64((Float64(beta + Float64(2.0 * i)) ^ 2.0) / (Float64(beta + i) ^ 2.0)))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 3.6e+206)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	else
		tmp = Float64(Float64(sqrt(i) / beta) * sqrt(Float64(alpha + i))) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (2.0 * i);
	t_1 = (0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25);
	tmp = 0.0;
	if (beta <= 1.32e+77)
		tmp = (0.0625 + ((0.25 * (t_1 / i)) - (((0.015625 * (-1.0 + ((beta + alpha) ^ 2.0))) + ((beta + alpha) * ((0.25 * t_1) - (0.0625 * (beta + alpha))))) / (i ^ 2.0)))) - (0.0625 * ((beta + alpha) / i));
	elseif (beta <= 1.8e+109)
		tmp = (i * (i / (((beta + (2.0 * i)) ^ 2.0) / ((beta + i) ^ 2.0)))) / (-1.0 + (t_0 * t_0));
	elseif (beta <= 3.6e+206)
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	else
		tmp = ((sqrt(i) / beta) * sqrt((alpha + i))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.25 * N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.32e+77], N[(N[(0.0625 + N[(N[(0.25 * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.015625 * N[(-1.0 + N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * N[(N[(0.25 * t$95$1), $MachinePrecision] - N[(0.0625 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.8e+109], N[(N[(i * N[(i / N[(N[Power[N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.6e+206], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Sqrt[i], $MachinePrecision] / beta), $MachinePrecision] * N[Sqrt[N[(alpha + i), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.32e77

   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. Taylor expanded in i around inf 35.0%

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

   3. Taylor expanded in i around inf 79.0%

      \[\leadsto \color{blue}{\left(0.0625 + \left(-1 \cdot \frac{0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right) + \left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right) - 0.0625 \cdot \left(\alpha + \beta\right)\right)}{{i}^{2}} + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right)\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   if 1.32e77 < beta < 1.8e109

   1. Initial program 31.4%

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

   3. Applied egg-rr 29.6%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 31.4%

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

      3. associate-*r* 53.2%

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

      4. associate-*l* 60.9%

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

      5. +-commutative 60.9%

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

      6. +-commutative 60.9%

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

      7. *-commutative 60.9%

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

      8. +-commutative 60.9%

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

      9. +-commutative 60.9%

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

      10. +-commutative 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in alpha around 0 53.5%

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

      1. associate-/l* 61.1%

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

      2. *-commutative 61.1%

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

      3. +-commutative 61.1%

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

   8. Simplified 61.1%

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

   if 1.8e109 < beta < 3.60000000000000028e206

   1. Initial program 0.4%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.4%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in i around inf 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. distribute-lft-out 63.0%

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

      3. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in alpha around 0 59.2%

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

      1. associate-*r/ 59.2%

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

      2. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in beta around 0 59.2%

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

   11. Taylor expanded in alpha around 0 63.0%

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

   if 3.60000000000000028e206 < 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. Taylor expanded in beta around inf 34.2%

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

   3. Taylor expanded in beta around inf 34.2%

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

      1. associate-/l* 35.8%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. +-commutative 35.8%

         \[\leadsto \frac{i}{\frac{{\beta}^{2}}{\color{blue}{i + \alpha}}} \]

   5. Simplified 35.8%

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

      1. add-sqr-sqrt 35.8%

         \[\leadsto \color{blue}{\sqrt{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \cdot \sqrt{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}}} \]

      2. pow2 35.8%

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

      3. associate-/r/ 35.8%

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

   7. Applied egg-rr 35.8%

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

      1. sqrt-prod 35.8%

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

      2. sqrt-div 35.8%

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

      3. unpow2 35.8%

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

      4. sqrt-prod 81.3%

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

      5. add-sqr-sqrt 81.3%

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

   9. Applied egg-rr 81.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+77}:\\ \;\;\;\;\left(0.0625 + \left(0.25 \cdot \frac{0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25}{i} - \frac{0.015625 \cdot \left(-1 + {\left(\beta + \alpha\right)}^{2}\right) + \left(\beta + \alpha\right) \cdot \left(0.25 \cdot \left(0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25\right) - 0.0625 \cdot \left(\beta + \alpha\right)\right)}{{i}^{2}}\right)\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+206}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\beta} \cdot \sqrt{\alpha + i}\right)}^{2}\\ \end{array} \]

Alternative 2: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := -1 + t_1\\ t_3 := i \cdot \left(\left(\beta + \alpha\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \beta \cdot \alpha\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = -1.0 + t_1;
	double t_3 = i * ((beta + alpha) + i);
	double tmp;
	if ((((t_3 * (t_3 + (beta * alpha))) / t_1) / t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (i * (i / (Math.pow((beta + (2.0 * i)), 2.0) / Math.pow((beta + i), 2.0)))) / t_2;
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = -1.0 + t_1
	t_3 = i * ((beta + alpha) + i)
	tmp = 0
	if (((t_3 * (t_3 + (beta * alpha))) / t_1) / t_2) <= math.inf:
		tmp = (i * (i / (math.pow((beta + (2.0 * i)), 2.0) / math.pow((beta + i), 2.0)))) / t_2
	else:
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)
	return tmp

Julia

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

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = -1.0 + t_1;
	t_3 = i * ((beta + alpha) + i);
	tmp = 0.0;
	if ((((t_3 * (t_3 + (beta * alpha))) / t_1) / t_2) <= Inf)
		tmp = (i * (i / (((beta + (2.0 * i)) ^ 2.0) / ((beta + i) ^ 2.0)))) / t_2;
	else
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.9%

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

   3. Applied egg-rr 42.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 42.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 44.7%

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

      3. associate-*r* 66.9%

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

      4. associate-*l* 99.3%

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

      5. +-commutative 99.3%

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

      6. +-commutative 99.3%

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

      7. *-commutative 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in alpha around 0 60.4%

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

      1. associate-/l* 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

   8. Simplified 91.1%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 75.7%

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

      1. cancel-sign-sub-inv 75.7%

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

      2. distribute-lft-out 75.7%

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

      3. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in alpha around 0 73.0%

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

      1. associate-*r/ 72.4%

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

      2. *-commutative 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in beta around 0 73.0%

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

   11. Taylor expanded in alpha around 0 74.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\beta + \alpha\right) + i\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \leq \infty:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 3: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{i \cdot \left(\left(\alpha + i\right) + \frac{\left(i \cdot \left(\alpha + i\right) + \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right) + -2 \cdot \left(\left(\alpha + i\right) \cdot \left(\alpha + 2 \cdot i\right)\right)}{\beta}\right)}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+206}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* 2.0 i))))
   (if (<= beta 1.1e+95)
     0.0625
     (if (<= beta 5.2e+108)
       (/
        (*
         i
         (+
          (+ alpha i)
          (/
           (+
            (+ (* i (+ alpha i)) (* (+ alpha i) (+ alpha i)))
            (* -2.0 (* (+ alpha i) (+ alpha (* 2.0 i)))))
           beta)))
        (+ -1.0 (* t_0 t_0)))
       (if (<= beta 3.6e+206)
         (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125))
         (pow (/ i beta) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double tmp;
	if (beta <= 1.1e+95) {
		tmp = 0.0625;
	} else if (beta <= 5.2e+108) {
		tmp = (i * ((alpha + i) + ((((i * (alpha + i)) + ((alpha + i) * (alpha + i))) + (-2.0 * ((alpha + i) * (alpha + (2.0 * i))))) / beta))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+206) {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	} else {
		tmp = pow((i / beta), 2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + (2.0d0 * i)
    if (beta <= 1.1d+95) then
        tmp = 0.0625d0
    else if (beta <= 5.2d+108) then
        tmp = (i * ((alpha + i) + ((((i * (alpha + i)) + ((alpha + i) * (alpha + i))) + ((-2.0d0) * ((alpha + i) * (alpha + (2.0d0 * i))))) / beta))) / ((-1.0d0) + (t_0 * t_0))
    else if (beta <= 3.6d+206) then
        tmp = (0.0625d0 + (0.125d0 * (beta / i))) + ((beta / i) * (-0.125d0))
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double tmp;
	if (beta <= 1.1e+95) {
		tmp = 0.0625;
	} else if (beta <= 5.2e+108) {
		tmp = (i * ((alpha + i) + ((((i * (alpha + i)) + ((alpha + i) * (alpha + i))) + (-2.0 * ((alpha + i) * (alpha + (2.0 * i))))) / beta))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+206) {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (2.0 * i)
	tmp = 0
	if beta <= 1.1e+95:
		tmp = 0.0625
	elif beta <= 5.2e+108:
		tmp = (i * ((alpha + i) + ((((i * (alpha + i)) + ((alpha + i) * (alpha + i))) + (-2.0 * ((alpha + i) * (alpha + (2.0 * i))))) / beta))) / (-1.0 + (t_0 * t_0))
	elif beta <= 3.6e+206:
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	tmp = 0.0
	if (beta <= 1.1e+95)
		tmp = 0.0625;
	elseif (beta <= 5.2e+108)
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) + Float64(Float64(Float64(Float64(i * Float64(alpha + i)) + Float64(Float64(alpha + i) * Float64(alpha + i))) + Float64(-2.0 * Float64(Float64(alpha + i) * Float64(alpha + Float64(2.0 * i))))) / beta))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 3.6e+206)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (2.0 * i);
	tmp = 0.0;
	if (beta <= 1.1e+95)
		tmp = 0.0625;
	elseif (beta <= 5.2e+108)
		tmp = (i * ((alpha + i) + ((((i * (alpha + i)) + ((alpha + i) * (alpha + i))) + (-2.0 * ((alpha + i) * (alpha + (2.0 * i))))) / beta))) / (-1.0 + (t_0 * t_0));
	elseif (beta <= 3.6e+206)
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.1e+95], 0.0625, If[LessEqual[beta, 5.2e+108], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] + N[(N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + i), $MachinePrecision] * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(alpha + i), $MachinePrecision] * N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.6e+206], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.0999999999999999e95

   1. Initial program 18.7%

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

      1. associate-/l/ 16.0%

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

      2. associate-*l* 16.0%

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

      3. times-frac 27.3%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in i around inf 81.5%

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

   if 1.0999999999999999e95 < beta < 5.2000000000000005e108

   1. Initial program 29.6%

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

   3. Applied egg-rr 28.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 28.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(i \cdot \left(\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(i + \left(\beta + \alpha\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 29.6%

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

      3. associate-*r* 56.8%

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

      4. associate-*l* 70.6%

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

      5. +-commutative 70.6%

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

      6. +-commutative 70.6%

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

      7. *-commutative 70.6%

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

      8. +-commutative 70.6%

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

      9. +-commutative 70.6%

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

      10. +-commutative 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in beta around -inf 57.1%

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

      1. mul-1-neg 57.1%

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

      2. unsub-neg 57.1%

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

      3. mul-1-neg 57.1%

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

      4. +-commutative 57.1%

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

      5. mul-1-neg 57.1%

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

      6. unsub-neg 57.1%

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

      7. mul-1-neg 57.1%

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

   8. Simplified 57.1%

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

   if 5.2000000000000005e108 < beta < 3.60000000000000028e206

   1. Initial program 0.4%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.4%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in i around inf 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. distribute-lft-out 63.0%

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

      3. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in alpha around 0 59.2%

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

      1. associate-*r/ 59.2%

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

      2. *-commutative 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in beta around 0 59.2%

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

   11. Taylor expanded in alpha around 0 63.0%

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

   if 3.60000000000000028e206 < 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. Taylor expanded in beta around inf 34.2%

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

   3. Taylor expanded in beta around inf 34.2%

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

      1. associate-/l* 35.8%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. +-commutative 35.8%

         \[\leadsto \frac{i}{\frac{{\beta}^{2}}{\color{blue}{i + \alpha}}} \]

   5. Simplified 35.8%

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

      1. add-sqr-sqrt 35.8%

         \[\leadsto \color{blue}{\sqrt{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \cdot \sqrt{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}}} \]

      2. pow2 35.8%

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

      3. associate-/r/ 35.8%

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

   7. Applied egg-rr 35.8%

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

   8. Taylor expanded in i around inf 77.5%

      \[\leadsto {\color{blue}{\left(\frac{i}{\beta}\right)}}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{i \cdot \left(\left(\alpha + i\right) + \frac{\left(i \cdot \left(\alpha + i\right) + \left(\alpha + i\right) \cdot \left(\alpha + i\right)\right) + -2 \cdot \left(\left(\alpha + i\right) \cdot \left(\alpha + 2 \cdot i\right)\right)}{\beta}\right)}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+206}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]

Alternative 4: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(\left(\beta + \alpha\right) + i\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \beta \cdot \alpha\right)}{t_1}}{-1 + t_1}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ beta alpha) i)))
        (t_3 (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1))))
   (if (<= t_3 0.1)
     t_3
     (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((beta + alpha) + i);
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (beta + alpha) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((beta + alpha) + i)
    t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / ((-1.0d0) + t_1)
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.125d0 * (beta / i))) + ((beta / i) * (-0.125d0))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((beta + alpha) + i);
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((beta + alpha) + i)
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(beta + alpha) + i))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((beta + alpha) + i);
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.6%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.6%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in i around inf 75.5%

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

      1. cancel-sign-sub-inv 75.5%

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

      2. distribute-lft-out 75.5%

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

      3. metadata-eval 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in alpha around 0 73.2%

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

      1. associate-*r/ 72.8%

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

      2. *-commutative 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in beta around 0 73.2%

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

   11. Taylor expanded in alpha around 0 74.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\beta + \alpha\right) + i\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\beta + \alpha\right) + i\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 5: 77.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
}

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 + (0.125d0 * (beta / i))) + ((beta / i) * (-0.125d0))
end function

Java

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

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125))
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
end

Wolfram

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

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 11.8%

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

   2. associate-*l* 11.8%

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

   3. times-frac 21.6%

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

3. Simplified 21.6%

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

4. Taylor expanded in i around inf 75.4%

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

   1. cancel-sign-sub-inv 75.4%

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

   2. distribute-lft-out 75.4%

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

   3. metadata-eval 75.4%

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

6. Simplified 75.4%

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

7. Taylor expanded in alpha around 0 73.4%

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

   1. associate-*r/ 73.0%

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

   2. *-commutative 73.0%

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

9. Simplified 73.0%

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

10. Taylor expanded in beta around 0 73.4%

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

11. Taylor expanded in alpha around 0 74.6%

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

12. Final simplification 74.6%

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

Alternative 6: 74.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+256}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+256) 0.0625 (/ (* (+ beta alpha) 0.0) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+256) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	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+256) then
        tmp = 0.0625d0
    else
        tmp = ((beta + alpha) * 0.0d0) / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+256) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1e+256:
		tmp = 0.0625
	else:
		tmp = ((beta + alpha) * 0.0) / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+256)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(beta + alpha) * 0.0) / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1e+256)
		tmp = 0.0625;
	else
		tmp = ((beta + alpha) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1e+256], 0.0625, N[(N[(N[(beta + alpha), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e256

   1. Initial program 15.3%

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

      1. associate-/l/ 12.4%

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

      2. associate-*l* 12.4%

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

      3. times-frac 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in i around inf 74.3%

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

   if 1e256 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 56.8%

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

      1. cancel-sign-sub-inv 56.8%

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

      2. distribute-lft-out 56.8%

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

      3. metadata-eval 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in i around 0 64.4%

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

      1. distribute-rgt-out 64.4%

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

      2. +-commutative 64.4%

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

      3. metadata-eval 64.4%

         \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{0}}{i} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\left(\beta + \alpha\right) \cdot 0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+256}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0}{i}\\ \end{array} \]

Alternative 7: 71.3% accurate, 53.0× 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 14.6%

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

   1. associate-/l/ 11.8%

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

   2. associate-*l* 11.8%

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

   3. times-frac 21.6%

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

3. Simplified 21.6%

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

4. Taylor expanded in i around inf 70.7%

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

5. Final simplification 70.7%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023321 
(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)))