Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 84.8%

Time: 17.5s

Alternatives: 10

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: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{i}{i + \alpha}\\ t_1 := \left(\alpha + \beta\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(4 \cdot t_0 + \left(2 \cdot \frac{\alpha}{i + \alpha} + \frac{\beta}{i + \alpha}\right)\right) - t_0}}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+196}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ i (+ i alpha))) (t_1 (+ (+ alpha beta) (* i 2.0))))
   (if (<= beta 2.75e+125)
     0.0625
     (if (<= beta 1.1e+154)
       (/
        (*
         i
         (/
          (+ i (+ alpha beta))
          (-
           (+
            (* 4.0 t_0)
            (+ (* 2.0 (/ alpha (+ i alpha))) (/ beta (+ i alpha))))
           t_0)))
        (+ (* t_1 t_1) -1.0))
       (if (<= beta 1.3e+196)
         (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125)))
         (* (/ (+ i alpha) beta) (/ i beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = i / (i + alpha);
	double t_1 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (beta <= 2.75e+125) {
		tmp = 0.0625;
	} else if (beta <= 1.1e+154) {
		tmp = (i * ((i + (alpha + beta)) / (((4.0 * t_0) + ((2.0 * (alpha / (i + alpha))) + (beta / (i + alpha)))) - t_0))) / ((t_1 * t_1) + -1.0);
	} else if (beta <= 1.3e+196) {
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 = i / (i + alpha)
    t_1 = (alpha + beta) + (i * 2.0d0)
    if (beta <= 2.75d+125) then
        tmp = 0.0625d0
    else if (beta <= 1.1d+154) then
        tmp = (i * ((i + (alpha + beta)) / (((4.0d0 * t_0) + ((2.0d0 * (alpha / (i + alpha))) + (beta / (i + alpha)))) - t_0))) / ((t_1 * t_1) + (-1.0d0))
    else if (beta <= 1.3d+196) then
        tmp = ((-0.125d0) * (beta / i)) + (0.0625d0 + ((beta / i) * 0.125d0))
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = i / (i + alpha);
	double t_1 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (beta <= 2.75e+125) {
		tmp = 0.0625;
	} else if (beta <= 1.1e+154) {
		tmp = (i * ((i + (alpha + beta)) / (((4.0 * t_0) + ((2.0 * (alpha / (i + alpha))) + (beta / (i + alpha)))) - t_0))) / ((t_1 * t_1) + -1.0);
	} else if (beta <= 1.3e+196) {
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = i / (i + alpha)
	t_1 = (alpha + beta) + (i * 2.0)
	tmp = 0
	if beta <= 2.75e+125:
		tmp = 0.0625
	elif beta <= 1.1e+154:
		tmp = (i * ((i + (alpha + beta)) / (((4.0 * t_0) + ((2.0 * (alpha / (i + alpha))) + (beta / (i + alpha)))) - t_0))) / ((t_1 * t_1) + -1.0)
	elif beta <= 1.3e+196:
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125))
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(i / Float64(i + alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 2.75e+125)
		tmp = 0.0625;
	elseif (beta <= 1.1e+154)
		tmp = Float64(Float64(i * Float64(Float64(i + Float64(alpha + beta)) / Float64(Float64(Float64(4.0 * t_0) + Float64(Float64(2.0 * Float64(alpha / Float64(i + alpha))) + Float64(beta / Float64(i + alpha)))) - t_0))) / Float64(Float64(t_1 * t_1) + -1.0));
	elseif (beta <= 1.3e+196)
		tmp = Float64(Float64(-0.125 * Float64(beta / i)) + Float64(0.0625 + Float64(Float64(beta / i) * 0.125)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = i / (i + alpha);
	t_1 = (alpha + beta) + (i * 2.0);
	tmp = 0.0;
	if (beta <= 2.75e+125)
		tmp = 0.0625;
	elseif (beta <= 1.1e+154)
		tmp = (i * ((i + (alpha + beta)) / (((4.0 * t_0) + ((2.0 * (alpha / (i + alpha))) + (beta / (i + alpha)))) - t_0))) / ((t_1 * t_1) + -1.0);
	elseif (beta <= 1.3e+196)
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i / N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.75e+125], 0.0625, If[LessEqual[beta, 1.1e+154], N[(N[(i * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * t$95$0), $MachinePrecision] + N[(N[(2.0 * N[(alpha / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.3e+196], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.74999999999999998e125

   1. Initial program 20.4%

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

      1. associate-/l/ 17.5%

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

      2. associate-*l* 17.4%

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

      3. times-frac 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in i around inf 82.3%

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

   if 2.74999999999999998e125 < beta < 1.1000000000000001e154

   1. Initial program 1.6%

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

      1. expm1-log1p-u 1.6%

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

      2. expm1-udef 1.6%

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

   3. Applied egg-rr 55.0%

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

      1. expm1-def 55.0%

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

      2. expm1-log1p 61.0%

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

      3. associate-*r/ 61.1%

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

      4. +-commutative 61.1%

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

      5. +-commutative 61.1%

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

      6. +-commutative 61.1%

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

      7. +-commutative 61.1%

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

      8. +-commutative 61.1%

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

      9. *-commutative 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in beta around inf 61.9%

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

   if 1.1000000000000001e154 < beta < 1.30000000000000006e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in i around inf 74.5%

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

      1. cancel-sign-sub-inv 74.5%

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

      2. distribute-lft-out 74.5%

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

      3. metadata-eval 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in alpha around 0 74.5%

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

   if 1.30000000000000006e196 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in beta around inf 10.5%

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

      1. *-commutative 10.5%

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

      2. associate-/l* 13.6%

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

      3. +-commutative 13.6%

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

      4. unpow2 13.6%

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

   6. Simplified 13.6%

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

   7. Taylor expanded in beta around 0 13.6%

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

      1. unpow2 13.6%

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

      2. associate-/l* 41.6%

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

   9. Simplified 41.6%

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

       1. associate-/r/ 71.9%

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

   11. Applied egg-rr 71.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(4 \cdot \frac{i}{i + \alpha} + \left(2 \cdot \frac{\alpha}{i + \alpha} + \frac{\beta}{i + \alpha}\right)\right) - \frac{i}{i + \alpha}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+196}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2: 84.1% accurate, 0.1× speedup

Math

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

FPCore

NOTE: alpha and beta 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 (+ t_1 -1.0))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3)))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       i
       (/
        t_3
        (/ (pow (fma i 2.0 (+ alpha beta)) 2.0) (fma i t_3 (* alpha beta)))))
      t_2)
     (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125))))))

C

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

Julia

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

Wolfram

NOTE: alpha and beta 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[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$3 / N[(N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.7%

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

      1. expm1-log1p-u 41.3%

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

      2. expm1-udef 41.3%

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

   3. Applied egg-rr 90.5%

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

      1. expm1-def 90.5%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. *-commutative 99.8%

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

   5. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in i around inf 77.7%

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

      1. cancel-sign-sub-inv 77.7%

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

      2. distribute-lft-out 77.7%

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

      3. metadata-eval 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in alpha around 0 70.4%

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

4. Final simplification 81.3%

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

Alternative 3: 84.1% accurate, 0.1× speedup

Math

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

FPCore

NOTE: alpha and beta 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 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (+ beta (+ i alpha))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       i
       (*
        (fma i t_4 (* alpha beta))
        (/ t_4 (pow (fma i 2.0 (+ alpha beta)) 2.0))))
      t_2)
     (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125))))))

C

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

Julia

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

Wolfram

NOTE: alpha and beta 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[(t$95$1 + -1.0), $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]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.7%

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

      1. expm1-log1p-u 41.3%

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

      2. expm1-udef 41.3%

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

   3. Applied egg-rr 90.5%

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

      1. expm1-def 90.5%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. *-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. associate-/r/ 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Simplified 99.7%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in i around inf 77.7%

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

      1. cancel-sign-sub-inv 77.7%

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

      2. distribute-lft-out 77.7%

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

      3. metadata-eval 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in alpha around 0 70.4%

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

4. Final simplification 81.2%

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

Alternative 4: 83.6% accurate, 0.3× speedup

Math

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

FPCore

NOTE: alpha and beta 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 (+ t_1 -1.0))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3)))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/ (* i (/ t_3 (/ (pow (+ beta (* i 2.0)) 2.0) (* i (+ i beta))))) t_2)
     (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha and beta 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[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$3 / N[(N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.7%

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

      1. expm1-log1p-u 41.3%

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

      2. expm1-udef 41.3%

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

   3. Applied egg-rr 90.5%

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

      1. expm1-def 90.5%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 95.1%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in i around inf 77.7%

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

      1. cancel-sign-sub-inv 77.7%

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

      2. distribute-lft-out 77.7%

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

      3. metadata-eval 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in alpha around 0 70.4%

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

4. Final simplification 79.6%

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

Alternative 5: 84.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.05 \cdot 10^{+126}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+196}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.05e+126)
   0.0625
   (if (<= beta 1.32e+154)
     (* (+ i alpha) (/ 1.0 (/ (* beta beta) i)))
     (if (<= beta 1.3e+196)
       (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125)))
       (* (/ (+ i alpha) beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.05e+126) {
		tmp = 0.0625;
	} else if (beta <= 1.32e+154) {
		tmp = (i + alpha) * (1.0 / ((beta * beta) / i));
	} else if (beta <= 1.3e+196) {
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 5.05d+126) then
        tmp = 0.0625d0
    else if (beta <= 1.32d+154) then
        tmp = (i + alpha) * (1.0d0 / ((beta * beta) / i))
    else if (beta <= 1.3d+196) then
        tmp = ((-0.125d0) * (beta / i)) + (0.0625d0 + ((beta / i) * 0.125d0))
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.05e+126) {
		tmp = 0.0625;
	} else if (beta <= 1.32e+154) {
		tmp = (i + alpha) * (1.0 / ((beta * beta) / i));
	} else if (beta <= 1.3e+196) {
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.05e+126:
		tmp = 0.0625
	elif beta <= 1.32e+154:
		tmp = (i + alpha) * (1.0 / ((beta * beta) / i))
	elif beta <= 1.3e+196:
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125))
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.05e+126)
		tmp = 0.0625;
	elseif (beta <= 1.32e+154)
		tmp = Float64(Float64(i + alpha) * Float64(1.0 / Float64(Float64(beta * beta) / i)));
	elseif (beta <= 1.3e+196)
		tmp = Float64(Float64(-0.125 * Float64(beta / i)) + Float64(0.0625 + Float64(Float64(beta / i) * 0.125)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.05e+126)
		tmp = 0.0625;
	elseif (beta <= 1.32e+154)
		tmp = (i + alpha) * (1.0 / ((beta * beta) / i));
	elseif (beta <= 1.3e+196)
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.05e+126], 0.0625, If[LessEqual[beta, 1.32e+154], N[(N[(i + alpha), $MachinePrecision] * N[(1.0 / N[(N[(beta * beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.3e+196], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 5.05000000000000026e126

   1. Initial program 20.4%

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

      1. associate-/l/ 17.5%

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

      2. associate-*l* 17.4%

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

      3. times-frac 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in i around inf 82.3%

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

   if 5.05000000000000026e126 < beta < 1.31999999999999998e154

   1. Initial program 1.6%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 1.6%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in beta around inf 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-/l* 58.0%

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

      3. +-commutative 58.0%

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

      4. unpow2 58.0%

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

   6. Simplified 58.0%

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

      1. div-inv 58.1%

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

   8. Applied egg-rr 58.1%

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

   if 1.31999999999999998e154 < beta < 1.30000000000000006e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in i around inf 74.5%

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

      1. cancel-sign-sub-inv 74.5%

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

      2. distribute-lft-out 74.5%

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

      3. metadata-eval 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in alpha around 0 74.5%

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

   if 1.30000000000000006e196 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in beta around inf 10.5%

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

      1. *-commutative 10.5%

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

      2. associate-/l* 13.6%

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

      3. +-commutative 13.6%

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

      4. unpow2 13.6%

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

   6. Simplified 13.6%

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

   7. Taylor expanded in beta around 0 13.6%

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

      1. unpow2 13.6%

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

      2. associate-/l* 41.6%

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

   9. Simplified 41.6%

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

       1. associate-/r/ 71.9%

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

   11. Applied egg-rr 71.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.05 \cdot 10^{+126}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+196}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6: 85.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.1e+126) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+126) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.1d+126) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+126) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.1e+126:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.1e+126)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.1e+126)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.1e+126], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.09999999999999999e126

   1. Initial program 20.4%

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

      1. associate-/l/ 17.5%

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

      2. associate-*l* 17.4%

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

      3. times-frac 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in i around inf 82.3%

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

   if 1.09999999999999999e126 < beta

   1. Initial program 0.4%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.4%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in beta around inf 20.5%

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

      1. *-commutative 20.5%

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

      2. associate-/l* 22.8%

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

      3. +-commutative 22.8%

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

      4. unpow2 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in beta around 0 22.8%

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

      1. unpow2 22.8%

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

      2. associate-/l* 40.1%

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

   9. Simplified 40.1%

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

       1. associate-/r/ 55.0%

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

   11. Applied egg-rr 55.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7: 74.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+263}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.4e+263) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+263) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 2.4d+263) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+263) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.4e+263:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.4e+263)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.4e+263)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.4e+263], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4e263

   1. Initial program 17.3%

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

      1. associate-/l/ 14.8%

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

      2. associate-*l* 14.7%

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

      3. times-frac 22.8%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in i around inf 76.4%

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

   if 2.4e263 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in beta around inf 20.8%

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

      1. *-commutative 20.8%

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

      2. associate-/l* 24.1%

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

      3. +-commutative 24.1%

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

      4. unpow2 24.1%

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

   6. Simplified 24.1%

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

   7. Taylor expanded in beta around 0 24.1%

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

      1. unpow2 24.1%

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

      2. associate-/l* 41.9%

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

   9. Simplified 41.9%

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

   10. Taylor expanded in alpha around inf 21.1%

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

       1. unpow2 21.1%

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

       2. times-frac 49.9%

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

   12. Simplified 49.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+263}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 8: 83.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.35e+196) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.35e+196) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.35d+196) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.35e+196) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.35e+196:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.35e+196)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.35e+196)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.35e+196], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.34999999999999998e196

   1. Initial program 18.3%

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

      1. associate-/l/ 15.6%

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

      2. associate-*l* 15.5%

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

      3. times-frac 24.1%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in i around inf 78.8%

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

   if 1.34999999999999998e196 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in beta around inf 10.5%

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

      1. *-commutative 10.5%

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

      2. associate-/l* 13.6%

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

      3. +-commutative 13.6%

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

      4. unpow2 13.6%

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

   6. Simplified 13.6%

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

      1. associate-/r/ 13.6%

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

   8. Applied egg-rr 13.6%

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

   9. Taylor expanded in alpha around 0 11.2%

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

       1. unpow2 11.2%

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

       2. unpow2 11.2%

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

       3. times-frac 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9: 74.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+268}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.4e+268) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+268) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 3.4d+268) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+268) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.4e+268:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.4e+268)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.4e+268)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.4e+268], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4000000000000003e268

   1. Initial program 17.3%

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

      1. associate-/l/ 14.7%

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

      2. associate-*l* 14.7%

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

      3. times-frac 22.7%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in i around inf 76.1%

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

   if 3.4000000000000003e268 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 35.8%

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

      1. cancel-sign-sub-inv 35.8%

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

      2. distribute-lft-out 35.8%

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

      3. metadata-eval 35.8%

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

   6. Applied egg-rr 35.8%

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

   7. Taylor expanded in i around 0 26.1%

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

      1. distribute-rgt-out 26.1%

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

      2. metadata-eval 26.1%

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

      3. mul0-rgt 26.1%

         \[\leadsto \frac{\color{blue}{0}}{i} \]

   9. Simplified 26.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+268}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 10: 71.5% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta 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;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

   1. associate-/l/ 14.1%

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

   2. associate-*l* 14.1%

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

   3. times-frac 21.8%

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

3. Simplified 39.8%

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

4. Taylor expanded in i around inf 73.7%

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

5. Final simplification 73.7%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023238 
(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)))