Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 82.5%

Time: 26.1s

Alternatives: 9

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: 15.9% 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: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \alpha + i \cdot 2\\ t_2 := 2 \cdot t\_1\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+165}:\\ \;\;\;\;i \cdot \left(\frac{\alpha + \left(i + \frac{i \cdot \left(i + \alpha\right) + -2 \cdot \left(\left(i + \alpha\right) \cdot t\_1\right)}{\beta}\right)}{\beta} \cdot \frac{1 + \frac{\left(\alpha + \left(i + \left(-2 \cdot \frac{t\_1 \cdot \left(\left(i + \alpha\right) - t\_2\right)}{\beta} - \frac{{t\_1}^{2}}{\beta}\right)\right)\right) - t\_2}{\beta}}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+244}:\\ \;\;\;\;\left(t\_0 + 0.0625\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))) (t_1 (+ alpha (* i 2.0))) (t_2 (* 2.0 t_1)))
   (if (<= beta 2.4e+136)
     0.0625
     (if (<= beta 9.5e+165)
       (*
        i
        (*
         (/
          (+
           alpha
           (+ i (/ (+ (* i (+ i alpha)) (* -2.0 (* (+ i alpha) t_1))) beta)))
          beta)
         (/
          (+
           1.0
           (/
            (-
             (+
              alpha
              (+
               i
               (-
                (* -2.0 (/ (* t_1 (- (+ i alpha) t_2)) beta))
                (/ (pow t_1 2.0) beta))))
             t_2)
            beta))
          beta)))
       (if (<= beta 6.2e+244)
         (- (+ t_0 0.0625) t_0)
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = alpha + (i * 2.0);
	double t_2 = 2.0 * t_1;
	double tmp;
	if (beta <= 2.4e+136) {
		tmp = 0.0625;
	} else if (beta <= 9.5e+165) {
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 + (((alpha + (i + ((-2.0 * ((t_1 * ((i + alpha) - t_2)) / beta)) - (pow(t_1, 2.0) / beta)))) - t_2) / beta)) / beta));
	} else if (beta <= 6.2e+244) {
		tmp = (t_0 + 0.0625) - t_0;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.125d0 * (beta / i)
    t_1 = alpha + (i * 2.0d0)
    t_2 = 2.0d0 * t_1
    if (beta <= 2.4d+136) then
        tmp = 0.0625d0
    else if (beta <= 9.5d+165) then
        tmp = i * (((alpha + (i + (((i * (i + alpha)) + ((-2.0d0) * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0d0 + (((alpha + (i + (((-2.0d0) * ((t_1 * ((i + alpha) - t_2)) / beta)) - ((t_1 ** 2.0d0) / beta)))) - t_2) / beta)) / beta))
    else if (beta <= 6.2d+244) then
        tmp = (t_0 + 0.0625d0) - t_0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = alpha + (i * 2.0);
	double t_2 = 2.0 * t_1;
	double tmp;
	if (beta <= 2.4e+136) {
		tmp = 0.0625;
	} else if (beta <= 9.5e+165) {
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 + (((alpha + (i + ((-2.0 * ((t_1 * ((i + alpha) - t_2)) / beta)) - (Math.pow(t_1, 2.0) / beta)))) - t_2) / beta)) / beta));
	} else if (beta <= 6.2e+244) {
		tmp = (t_0 + 0.0625) - t_0;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	t_1 = alpha + (i * 2.0)
	t_2 = 2.0 * t_1
	tmp = 0
	if beta <= 2.4e+136:
		tmp = 0.0625
	elif beta <= 9.5e+165:
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 + (((alpha + (i + ((-2.0 * ((t_1 * ((i + alpha) - t_2)) / beta)) - (math.pow(t_1, 2.0) / beta)))) - t_2) / beta)) / beta))
	elif beta <= 6.2e+244:
		tmp = (t_0 + 0.0625) - t_0
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	t_1 = Float64(alpha + Float64(i * 2.0))
	t_2 = Float64(2.0 * t_1)
	tmp = 0.0
	if (beta <= 2.4e+136)
		tmp = 0.0625;
	elseif (beta <= 9.5e+165)
		tmp = Float64(i * Float64(Float64(Float64(alpha + Float64(i + Float64(Float64(Float64(i * Float64(i + alpha)) + Float64(-2.0 * Float64(Float64(i + alpha) * t_1))) / beta))) / beta) * Float64(Float64(1.0 + Float64(Float64(Float64(alpha + Float64(i + Float64(Float64(-2.0 * Float64(Float64(t_1 * Float64(Float64(i + alpha) - t_2)) / beta)) - Float64((t_1 ^ 2.0) / beta)))) - t_2) / beta)) / beta)));
	elseif (beta <= 6.2e+244)
		tmp = Float64(Float64(t_0 + 0.0625) - t_0);
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	t_1 = alpha + (i * 2.0);
	t_2 = 2.0 * t_1;
	tmp = 0.0;
	if (beta <= 2.4e+136)
		tmp = 0.0625;
	elseif (beta <= 9.5e+165)
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 + (((alpha + (i + ((-2.0 * ((t_1 * ((i + alpha) - t_2)) / beta)) - ((t_1 ^ 2.0) / beta)))) - t_2) / beta)) / beta));
	elseif (beta <= 6.2e+244)
		tmp = (t_0 + 0.0625) - t_0;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * t$95$1), $MachinePrecision]}, If[LessEqual[beta, 2.4e+136], 0.0625, If[LessEqual[beta, 9.5e+165], N[(i * N[(N[(N[(alpha + N[(i + N[(N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(i + alpha), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(alpha + N[(i + N[(N[(-2.0 * N[(N[(t$95$1 * N[(N[(i + alpha), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.2e+244], N[(N[(t$95$0 + 0.0625), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.4e136

   1. Initial program 20.7%

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

      1. associate-/l/ 17.4%

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

      2. associate-*l* 17.3%

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

      3. associate-/l* 17.5%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in i around inf 83.0%

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

   if 2.4e136 < beta < 9.50000000000000017e165

   1. Initial program 9.9%

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

      1. associate-/l/ 0.7%

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

      2. associate-*l* 0.7%

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

      3. associate-/l* 0.7%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in beta around -inf 41.4%

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

   6. Taylor expanded in beta around -inf 41.1%

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

   if 9.50000000000000017e165 < beta < 6.20000000000000001e244

   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. associate-/l* 0.0%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in i around inf 61.9%

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

      1. sub-neg 61.9%

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

      2. +-commutative 61.9%

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

      3. fma-define 61.9%

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

      4. distribute-lft-out 61.9%

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

      5. associate-*r/ 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. unsub-neg 61.9%

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

      2. fma-undefine 61.9%

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

      3. associate-/l* 61.9%

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

      4. associate-*r* 62.1%

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

      5. metadata-eval 62.1%

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

      6. associate-/l* 62.1%

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

      7. associate-/l* 62.1%

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

   9. Simplified 62.1%

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

   10. Taylor expanded in alpha around 0 57.4%

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

   11. Taylor expanded in alpha around 0 62.1%

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

   if 6.20000000000000001e244 < 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. times-frac 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in beta around inf 16.0%

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

   6. Taylor expanded in beta around inf 95.1%

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

      1. +-commutative 95.1%

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

   8. Simplified 95.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+165}:\\ \;\;\;\;i \cdot \left(\frac{\alpha + \left(i + \frac{i \cdot \left(i + \alpha\right) + -2 \cdot \left(\left(i + \alpha\right) \cdot \left(\alpha + i \cdot 2\right)\right)}{\beta}\right)}{\beta} \cdot \frac{1 + \frac{\left(\alpha + \left(i + \left(-2 \cdot \frac{\left(\alpha + i \cdot 2\right) \cdot \left(\left(i + \alpha\right) - 2 \cdot \left(\alpha + i \cdot 2\right)\right)}{\beta} - \frac{{\left(\alpha + i \cdot 2\right)}^{2}}{\beta}\right)\right)\right) - 2 \cdot \left(\alpha + i \cdot 2\right)}{\beta}}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+244}:\\ \;\;\;\;\left(0.125 \cdot \frac{\beta}{i} + 0.0625\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$4), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $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 52.4%

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

      1. associate-/l/ 42.9%

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

      2. times-frac 99.5%

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

   3. Simplified 99.5%

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

   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. associate-/l* 0.0%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in i around inf 80.9%

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

      1. sub-neg 80.9%

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

      2. +-commutative 80.9%

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

      3. fma-define 80.9%

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

      4. distribute-lft-out 80.9%

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

      5. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. unsub-neg 80.9%

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

      2. fma-undefine 80.9%

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

      3. associate-/l* 80.9%

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

      4. associate-*r* 80.9%

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

      5. metadata-eval 80.9%

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

      6. associate-/l* 80.9%

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

      7. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in alpha around 0 78.3%

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

   11. Taylor expanded in alpha around 0 79.2%

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

4. Final simplification 85.7%

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

Alternative 3: 83.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(i * N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $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 52.4%

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

      1. associate-/l/ 42.9%

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

      2. associate-*l* 42.8%

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

      3. associate-/l* 43.3%

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

   3. Simplified 99.2%

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

   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. associate-/l* 0.0%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in i around inf 80.9%

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

      1. sub-neg 80.9%

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

      2. +-commutative 80.9%

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

      3. fma-define 80.9%

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

      4. distribute-lft-out 80.9%

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

      5. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. unsub-neg 80.9%

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

      2. fma-undefine 80.9%

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

      3. associate-/l* 80.9%

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

      4. associate-*r* 80.9%

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

      5. metadata-eval 80.9%

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

      6. associate-/l* 80.9%

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

      7. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in alpha around 0 78.3%

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

   11. Taylor expanded in alpha around 0 79.2%

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

4. Final simplification 85.6%

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

Alternative 4: 83.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + (alpha + beta));
	double t_1 = (alpha + beta) + (i * 2.0);
	double t_2 = t_1 * t_1;
	double t_3 = t_2 + -1.0;
	double t_4 = 0.125 * (beta / i);
	double tmp;
	if ((((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow(i, 2.0) * (Math.pow((i + beta), 2.0) / Math.pow((beta + (i * 2.0)), 2.0))) / t_3;
	} else {
		tmp = (t_4 + 0.0625) - t_4;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = i * (i + (alpha + beta))
	t_1 = (alpha + beta) + (i * 2.0)
	t_2 = t_1 * t_1
	t_3 = t_2 + -1.0
	t_4 = 0.125 * (beta / i)
	tmp = 0
	if (((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= math.inf:
		tmp = (math.pow(i, 2.0) * (math.pow((i + beta), 2.0) / math.pow((beta + (i * 2.0)), 2.0))) / t_3
	else:
		tmp = (t_4 + 0.0625) - t_4
	return tmp

Julia

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

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + (alpha + beta));
	t_1 = (alpha + beta) + (i * 2.0);
	t_2 = t_1 * t_1;
	t_3 = t_2 + -1.0;
	t_4 = 0.125 * (beta / i);
	tmp = 0.0;
	if ((((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= Inf)
		tmp = ((i ^ 2.0) * (((i + beta) ^ 2.0) / ((beta + (i * 2.0)) ^ 2.0))) / t_3;
	else
		tmp = (t_4 + 0.0625) - t_4;
	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[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$0 * N[(t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], Infinity], N[(N[(N[Power[i, 2.0], $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(t$95$4 + 0.0625), $MachinePrecision] - t$95$4), $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 52.4%

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

   3. Taylor expanded in alpha around 0 48.0%

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

      1. associate-/l* 90.7%

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

   5. Simplified 90.7%

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

   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. associate-/l* 0.0%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in i around inf 80.9%

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

      1. sub-neg 80.9%

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

      2. +-commutative 80.9%

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

      3. fma-define 80.9%

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

      4. distribute-lft-out 80.9%

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

      5. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. unsub-neg 80.9%

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

      2. fma-undefine 80.9%

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

      3. associate-/l* 80.9%

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

      4. associate-*r* 80.9%

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

      5. metadata-eval 80.9%

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

      6. associate-/l* 80.9%

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

      7. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in alpha around 0 78.3%

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

   11. Taylor expanded in alpha around 0 79.2%

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

4. Final simplification 82.9%

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

Alternative 5: 83.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = 0.125 * (beta / i);
	double t_3 = i * (i + (alpha + beta));
	double t_4 = Math.pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = i * (((i * (i + beta)) / (-1.0 + t_4)) * ((i + beta) / t_4));
	} else {
		tmp = (t_2 + 0.0625) - t_2;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = 0.125 * (beta / i)
	t_3 = i * (i + (alpha + beta))
	t_4 = math.pow((beta + (i * 2.0)), 2.0)
	tmp = 0
	if (((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= math.inf:
		tmp = i * (((i * (i + beta)) / (-1.0 + t_4)) * ((i + beta) / t_4))
	else:
		tmp = (t_2 + 0.0625) - t_2
	return tmp

Julia

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

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = 0.125 * (beta / i);
	t_3 = i * (i + (alpha + beta));
	t_4 = (beta + (i * 2.0)) ^ 2.0;
	tmp = 0.0;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Inf)
		tmp = i * (((i * (i + beta)) / (-1.0 + t_4)) * ((i + beta) / t_4));
	else
		tmp = (t_2 + 0.0625) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(i * N[(N[(N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $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 52.4%

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

      1. associate-/l/ 42.9%

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

      2. associate-*l* 42.8%

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

      3. associate-/l* 43.3%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in alpha around 0 89.2%

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

   6. Taylor expanded in alpha around 0 89.0%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 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. associate-/l* 0.0%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in i around inf 80.9%

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

      1. sub-neg 80.9%

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

      2. +-commutative 80.9%

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

      3. fma-define 80.9%

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

      4. distribute-lft-out 80.9%

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

      5. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. unsub-neg 80.9%

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

      2. fma-undefine 80.9%

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

      3. associate-/l* 80.9%

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

      4. associate-*r* 80.9%

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

      5. metadata-eval 80.9%

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

      6. associate-/l* 80.9%

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

      7. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in alpha around 0 78.3%

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

   11. Taylor expanded in alpha around 0 79.2%

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

4. Final simplification 82.3%

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

Alternative 6: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \alpha + i \cdot 2\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+167}:\\ \;\;\;\;i \cdot \left(\frac{\alpha + \left(i + \frac{i \cdot \left(i + \alpha\right) + -2 \cdot \left(\left(i + \alpha\right) \cdot t\_1\right)}{\beta}\right)}{\beta} \cdot \frac{1 - \frac{2 \cdot t\_1 - \left(i + \alpha\right)}{\beta}}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+245}:\\ \;\;\;\;\left(t\_0 + 0.0625\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))) (t_1 (+ alpha (* i 2.0))))
   (if (<= beta 2.6e+136)
     0.0625
     (if (<= beta 3.1e+167)
       (*
        i
        (*
         (/
          (+
           alpha
           (+ i (/ (+ (* i (+ i alpha)) (* -2.0 (* (+ i alpha) t_1))) beta)))
          beta)
         (/ (- 1.0 (/ (- (* 2.0 t_1) (+ i alpha)) beta)) beta)))
       (if (<= beta 3.2e+245)
         (- (+ t_0 0.0625) t_0)
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = alpha + (i * 2.0);
	double tmp;
	if (beta <= 2.6e+136) {
		tmp = 0.0625;
	} else if (beta <= 3.1e+167) {
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 - (((2.0 * t_1) - (i + alpha)) / beta)) / beta));
	} else if (beta <= 3.2e+245) {
		tmp = (t_0 + 0.0625) - t_0;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.125d0 * (beta / i)
    t_1 = alpha + (i * 2.0d0)
    if (beta <= 2.6d+136) then
        tmp = 0.0625d0
    else if (beta <= 3.1d+167) then
        tmp = i * (((alpha + (i + (((i * (i + alpha)) + ((-2.0d0) * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0d0 - (((2.0d0 * t_1) - (i + alpha)) / beta)) / beta))
    else if (beta <= 3.2d+245) then
        tmp = (t_0 + 0.0625d0) - t_0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = alpha + (i * 2.0);
	double tmp;
	if (beta <= 2.6e+136) {
		tmp = 0.0625;
	} else if (beta <= 3.1e+167) {
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 - (((2.0 * t_1) - (i + alpha)) / beta)) / beta));
	} else if (beta <= 3.2e+245) {
		tmp = (t_0 + 0.0625) - t_0;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	t_1 = alpha + (i * 2.0)
	tmp = 0
	if beta <= 2.6e+136:
		tmp = 0.0625
	elif beta <= 3.1e+167:
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 - (((2.0 * t_1) - (i + alpha)) / beta)) / beta))
	elif beta <= 3.2e+245:
		tmp = (t_0 + 0.0625) - t_0
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	t_1 = Float64(alpha + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 2.6e+136)
		tmp = 0.0625;
	elseif (beta <= 3.1e+167)
		tmp = Float64(i * Float64(Float64(Float64(alpha + Float64(i + Float64(Float64(Float64(i * Float64(i + alpha)) + Float64(-2.0 * Float64(Float64(i + alpha) * t_1))) / beta))) / beta) * Float64(Float64(1.0 - Float64(Float64(Float64(2.0 * t_1) - Float64(i + alpha)) / beta)) / beta)));
	elseif (beta <= 3.2e+245)
		tmp = Float64(Float64(t_0 + 0.0625) - t_0);
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	t_1 = alpha + (i * 2.0);
	tmp = 0.0;
	if (beta <= 2.6e+136)
		tmp = 0.0625;
	elseif (beta <= 3.1e+167)
		tmp = i * (((alpha + (i + (((i * (i + alpha)) + (-2.0 * ((i + alpha) * t_1))) / beta))) / beta) * ((1.0 - (((2.0 * t_1) - (i + alpha)) / beta)) / beta));
	elseif (beta <= 3.2e+245)
		tmp = (t_0 + 0.0625) - t_0;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6e+136], 0.0625, If[LessEqual[beta, 3.1e+167], N[(i * N[(N[(N[(alpha + N[(i + N[(N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(i + alpha), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(2.0 * t$95$1), $MachinePrecision] - N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.2e+245], N[(N[(t$95$0 + 0.0625), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.6000000000000001e136

   1. Initial program 20.7%

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

      1. associate-/l/ 17.4%

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

      2. associate-*l* 17.3%

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

      3. associate-/l* 17.5%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in i around inf 83.0%

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

   if 2.6000000000000001e136 < beta < 3.1e167

   1. Initial program 9.1%

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

      1. associate-/l/ 0.6%

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

      2. associate-*l* 0.6%

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

      3. associate-/l* 0.6%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in beta around -inf 37.9%

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

   6. Taylor expanded in beta around -inf 37.5%

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

   if 3.1e167 < beta < 3.20000000000000024e245

   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. associate-/l* 0.0%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in i around inf 60.0%

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

      1. sub-neg 60.0%

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

      2. +-commutative 60.0%

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

      3. fma-define 60.0%

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

      4. distribute-lft-out 60.0%

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

      5. associate-*r/ 60.0%

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

   7. Applied egg-rr 60.0%

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

      1. unsub-neg 60.0%

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

      2. fma-undefine 60.0%

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

      3. associate-/l* 60.0%

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

      4. associate-*r* 60.3%

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

      5. metadata-eval 60.3%

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

      6. associate-/l* 60.3%

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

      7. associate-/l* 60.3%

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

   9. Simplified 60.3%

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

   10. Taylor expanded in alpha around 0 60.1%

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

   11. Taylor expanded in alpha around 0 60.3%

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

   if 3.20000000000000024e245 < 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. times-frac 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in beta around inf 16.0%

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

   6. Taylor expanded in beta around inf 95.1%

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

      1. +-commutative 95.1%

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

   8. Simplified 95.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+167}:\\ \;\;\;\;i \cdot \left(\frac{\alpha + \left(i + \frac{i \cdot \left(i + \alpha\right) + -2 \cdot \left(\left(i + \alpha\right) \cdot \left(\alpha + i \cdot 2\right)\right)}{\beta}\right)}{\beta} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + i \cdot 2\right) - \left(i + \alpha\right)}{\beta}}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+245}:\\ \;\;\;\;\left(0.125 \cdot \frac{\beta}{i} + 0.0625\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+149} \lor \neg \left(\beta \leq 9.8 \cdot 10^{+244}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + 0.0625\right) - t\_0\\ \end{array} \end{array} \]

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 (* 0.125 (/ beta i))))
   (if (<= beta 2e+136)
     0.0625
     (if (or (<= beta 1.85e+149) (not (<= beta 9.8e+244)))
       (* (/ i beta) (/ (+ i alpha) beta))
       (- (+ t_0 0.0625) t_0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 2e+136) {
		tmp = 0.0625;
	} else if ((beta <= 1.85e+149) || !(beta <= 9.8e+244)) {
		tmp = (i / beta) * ((i + alpha) / beta);
	} else {
		tmp = (t_0 + 0.0625) - t_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 = 0.125d0 * (beta / i)
    if (beta <= 2d+136) then
        tmp = 0.0625d0
    else if ((beta <= 1.85d+149) .or. (.not. (beta <= 9.8d+244))) then
        tmp = (i / beta) * ((i + alpha) / beta)
    else
        tmp = (t_0 + 0.0625d0) - t_0
    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 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 2e+136) {
		tmp = 0.0625;
	} else if ((beta <= 1.85e+149) || !(beta <= 9.8e+244)) {
		tmp = (i / beta) * ((i + alpha) / beta);
	} else {
		tmp = (t_0 + 0.0625) - t_0;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	tmp = 0
	if beta <= 2e+136:
		tmp = 0.0625
	elif (beta <= 1.85e+149) or not (beta <= 9.8e+244):
		tmp = (i / beta) * ((i + alpha) / beta)
	else:
		tmp = (t_0 + 0.0625) - t_0
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (beta <= 2e+136)
		tmp = 0.0625;
	elseif ((beta <= 1.85e+149) || !(beta <= 9.8e+244))
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	else
		tmp = Float64(Float64(t_0 + 0.0625) - t_0);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = 0.0;
	if (beta <= 2e+136)
		tmp = 0.0625;
	elseif ((beta <= 1.85e+149) || ~((beta <= 9.8e+244)))
		tmp = (i / beta) * ((i + alpha) / beta);
	else
		tmp = (t_0 + 0.0625) - t_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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+136], 0.0625, If[Or[LessEqual[beta, 1.85e+149], N[Not[LessEqual[beta, 9.8e+244]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + 0.0625), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.00000000000000012e136

   1. Initial program 20.7%

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

      1. associate-/l/ 17.4%

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

      2. associate-*l* 17.3%

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

      3. associate-/l* 17.5%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in i around inf 83.0%

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

   if 2.00000000000000012e136 < beta < 1.84999999999999989e149 or 9.8e244 < beta

   1. Initial program 4.1%

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

      1. associate-/l/ 0.3%

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

      2. times-frac 26.7%

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

   3. Simplified 26.8%

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

   5. Taylor expanded in beta around inf 25.9%

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

   6. Taylor expanded in beta around inf 89.8%

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

      1. +-commutative 89.8%

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

   8. Simplified 89.8%

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

   if 1.84999999999999989e149 < beta < 9.8e244

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l* 0.0%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in i around inf 66.6%

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

      1. sub-neg 66.6%

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

      2. +-commutative 66.6%

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

      3. fma-define 66.6%

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

      4. distribute-lft-out 66.6%

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

      5. associate-*r/ 66.6%

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

   7. Applied egg-rr 66.6%

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

      1. unsub-neg 66.6%

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

      2. fma-undefine 66.6%

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

      3. associate-/l* 66.6%

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

      4. associate-*r* 66.8%

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

      5. metadata-eval 66.8%

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

      6. associate-/l* 66.8%

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

      7. associate-/l* 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in alpha around 0 63.1%

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

   11. Taylor expanded in alpha around 0 66.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+149} \lor \neg \left(\beta \leq 9.8 \cdot 10^{+244}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.125 \cdot \frac{\beta}{i} + 0.0625\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+149} \lor \neg \left(\beta \leq 2.3 \cdot 10^{+205}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.6e+136)
   0.0625
   (if (or (<= beta 2.3e+149) (not (<= beta 2.3e+205)))
     (* (/ i beta) (/ (+ i alpha) beta))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+136) {
		tmp = 0.0625;
	} else if ((beta <= 2.3e+149) || !(beta <= 2.3e+205)) {
		tmp = (i / beta) * ((i + alpha) / beta);
	} else {
		tmp = 0.0625;
	}
	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 <= 2.6d+136) then
        tmp = 0.0625d0
    else if ((beta <= 2.3d+149) .or. (.not. (beta <= 2.3d+205))) then
        tmp = (i / beta) * ((i + alpha) / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+136) {
		tmp = 0.0625;
	} else if ((beta <= 2.3e+149) || !(beta <= 2.3e+205)) {
		tmp = (i / beta) * ((i + alpha) / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.6e+136:
		tmp = 0.0625
	elif (beta <= 2.3e+149) or not (beta <= 2.3e+205):
		tmp = (i / beta) * ((i + alpha) / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.6e+136)
		tmp = 0.0625;
	elseif ((beta <= 2.3e+149) || !(beta <= 2.3e+205))
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.6e+136)
		tmp = 0.0625;
	elseif ((beta <= 2.3e+149) || ~((beta <= 2.3e+205)))
		tmp = (i / beta) * ((i + alpha) / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.6e+136], 0.0625, If[Or[LessEqual[beta, 2.3e+149], N[Not[LessEqual[beta, 2.3e+205]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6000000000000001e136 or 2.2999999999999998e149 < beta < 2.30000000000000007e205

   1. Initial program 19.2%

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

      1. associate-/l/ 16.1%

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

      2. associate-*l* 16.1%

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

      3. associate-/l* 16.2%

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

   3. Simplified 35.7%

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

   5. Taylor expanded in i around inf 82.1%

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

   if 2.6000000000000001e136 < beta < 2.2999999999999998e149 or 2.30000000000000007e205 < beta

   1. Initial program 2.8%

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

      1. associate-/l/ 0.2%

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

      2. times-frac 26.5%

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

   3. Simplified 26.5%

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

   5. Taylor expanded in beta around inf 28.7%

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

   6. Taylor expanded in beta around inf 82.9%

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

      1. +-commutative 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+149} \lor \neg \left(\beta \leq 2.3 \cdot 10^{+205}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% 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 16.8%

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

   1. associate-/l/ 13.8%

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

   2. associate-*l* 13.7%

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

   3. associate-/l* 13.9%

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

3. Simplified 34.3%

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

5. Taylor expanded in i around inf 72.4%

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

6. Final simplification 72.4%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))