Octave 3.8, jcobi/4

Percentage Accurate: 16.8% → 83.5%

Time: 19.0s

Alternatives: 14

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.8% 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: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_1 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_1}{i}\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* 2.0 i)))
        (t_1 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 6.6e+83)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_1)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_1 i))
     (if (<= beta 7.5e+153)
       (/
        (/ (* i i) (/ (pow (fma i 2.0 beta) 2.0) (pow (+ beta i) 2.0)))
        (+ -1.0 (* t_0 t_0)))
       (if (<= beta 9e+165)
         (- (+ 0.0625 (* 0.125 (/ beta i))) (* 0.125 (/ (+ beta alpha) i)))
         (/ (/ i (/ beta (+ alpha i))) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 6.6e+83) {
		tmp = (0.0625 + ((((beta + alpha) * t_1) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_1 / i);
	} else if (beta <= 7.5e+153) {
		tmp = ((i * i) / (pow(fma(i, 2.0, beta), 2.0) / pow((beta + i), 2.0))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 9e+165) {
		tmp = (0.0625 + (0.125 * (beta / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	t_1 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 6.6e+83)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_1) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_1 / i));
	elseif (beta <= 7.5e+153)
		tmp = Float64(Float64(Float64(i * i) / Float64((fma(i, 2.0, beta) ^ 2.0) / (Float64(beta + i) ^ 2.0))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 9e+165)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.6e+83], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7.5e+153], N[(N[(N[(i * i), $MachinePrecision] / N[(N[Power[N[(i * 2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9e+165], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 6.59999999999999969e83

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 6.59999999999999969e83 < beta < 7.50000000000000065e153

   1. Initial program 14.3%

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

   2. Taylor expanded in alpha around 0 14.4%

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

      1. associate-/l* 65.2%

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

      2. unpow2 65.2%

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

      3. +-commutative 65.2%

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

      4. *-commutative 65.2%

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

      5. fma-udef 65.2%

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

   4. Simplified 65.2%

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

   if 7.50000000000000065e153 < beta < 8.9999999999999993e165

   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 2.4%

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

   4. Taylor expanded in i around inf 77.4%

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

   5. Taylor expanded in alpha around 0 77.4%

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

   6. Taylor expanded in i around 0 77.4%

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

   if 8.9999999999999993e165 < 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 9.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 30.7%

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

      2. unpow2 30.7%

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

   6. Simplified 30.7%

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

      1. add-cbrt-cube 30.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 42.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 42.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 47.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 47.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 47.4%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 89.7%

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

       3. associate-*l/ 89.8%

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

       4. +-commutative 89.8%

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

   12. Applied egg-rr 89.8%

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

   13. Taylor expanded in beta around 0 67.6%

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

       1. associate-/l* 89.8%

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

   15. Simplified 89.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 2: 83.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+81}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_2 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_2}{i}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_0, t_0, -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}{t_0} \cdot \frac{t_1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (fma i 2.0 beta)))
        (t_1 (+ (+ beta alpha) i))
        (t_2 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 3.3e+81)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_2)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_2 i))
     (if (<= beta 6.5e+142)
       (*
        (/ i (fma t_0 t_0 -1.0))
        (* (/ (fma i t_1 (* beta alpha)) t_0) (/ t_1 t_0)))
       (/ (/ i (/ beta (+ alpha i))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + fma(i, 2.0, beta);
	double t_1 = (beta + alpha) + i;
	double t_2 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 3.3e+81) {
		tmp = (0.0625 + ((((beta + alpha) * t_2) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_2 / i);
	} else if (beta <= 6.5e+142) {
		tmp = (i / fma(t_0, t_0, -1.0)) * ((fma(i, t_1, (beta * alpha)) / t_0) * (t_1 / t_0));
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	t_1 = Float64(Float64(beta + alpha) + i)
	t_2 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 3.3e+81)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_2) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_2 / i));
	elseif (beta <= 6.5e+142)
		tmp = Float64(Float64(i / fma(t_0, t_0, -1.0)) * Float64(Float64(fma(i, t_1, Float64(beta * alpha)) / t_0) * Float64(t_1 / t_0)));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.3e+81], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.5e+142], N[(N[(i / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.3e81

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 3.3e81 < beta < 6.4999999999999997e142

   1. Initial program 17.1%

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

      1. associate-/l/ 0.7%

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

      2. associate-*l* 0.7%

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

      3. times-frac 37.4%

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

   3. Simplified 68.1%

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

   if 6.4999999999999997e142 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in beta around inf 29.1%

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

      1. associate-/l* 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

      1. add-cbrt-cube 28.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.1%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.0%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 37.8%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 37.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 41.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 41.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 41.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 80.9%

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

       3. associate-*l/ 81.0%

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

       4. +-commutative 81.0%

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

   12. Applied egg-rr 81.0%

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

   13. Taylor expanded in beta around 0 62.6%

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

       1. associate-/l* 80.9%

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

   15. Simplified 80.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+81}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\left(\beta + \alpha\right) + i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 3: 83.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+80}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_2 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_2}{i}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{i}{-1 + {t_0}^{2}} \cdot \frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}{t_0}\right) \cdot \frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha)))
        (t_1 (+ (+ beta alpha) i))
        (t_2 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 9.8e+80)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_2)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_2 i))
     (if (<= beta 3e+143)
       (*
        (* (/ i (+ -1.0 (pow t_0 2.0))) (/ (fma i t_1 (* beta alpha)) t_0))
        (/ t_1 t_0))
       (/ (/ i (/ beta (+ alpha i))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double t_2 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 9.8e+80) {
		tmp = (0.0625 + ((((beta + alpha) * t_2) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_2 / i);
	} else if (beta <= 3e+143) {
		tmp = ((i / (-1.0 + pow(t_0, 2.0))) * (fma(i, t_1, (beta * alpha)) / t_0)) * (t_1 / t_0);
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	t_2 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 9.8e+80)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_2) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_2 / i));
	elseif (beta <= 3e+143)
		tmp = Float64(Float64(Float64(i / Float64(-1.0 + (t_0 ^ 2.0))) * Float64(fma(i, t_1, Float64(beta * alpha)) / t_0)) * Float64(t_1 / t_0));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.8e+80], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+143], N[(N[(N[(i / N[(-1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9.79999999999999919e80

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 9.79999999999999919e80 < beta < 3.0000000000000001e143

   1. Initial program 17.1%

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

   3. Simplified 0.7%

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

   5. Applied egg-rr 68.1%

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

   if 3.0000000000000001e143 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in beta around inf 29.1%

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

      1. associate-/l* 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

      1. add-cbrt-cube 28.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.1%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.0%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 37.8%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 37.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 41.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 41.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 41.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 80.9%

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

       3. associate-*l/ 81.0%

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

       4. +-commutative 81.0%

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

   12. Applied egg-rr 81.0%

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

   13. Taylor expanded in beta around 0 62.6%

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

       1. associate-/l* 80.9%

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

   15. Simplified 80.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+80}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+143}:\\ \;\;\;\;\left(\frac{i}{-1 + {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\left(\beta + \alpha\right) + i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 4: 83.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_2 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_2}{i}\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+143}:\\ \;\;\;\;\frac{i}{\frac{-1 + t_0}{t_1 \cdot \frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (pow (fma i 2.0 (+ beta alpha)) 2.0))
        (t_1 (+ (+ beta alpha) i))
        (t_2 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 2.45e+83)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_2)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_2 i))
     (if (<= beta 1.85e+143)
       (/ i (/ (+ -1.0 t_0) (* t_1 (/ (fma i t_1 (* beta alpha)) t_0))))
       (/ (/ i (/ beta (+ alpha i))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = pow(fma(i, 2.0, (beta + alpha)), 2.0);
	double t_1 = (beta + alpha) + i;
	double t_2 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 2.45e+83) {
		tmp = (0.0625 + ((((beta + alpha) * t_2) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_2 / i);
	} else if (beta <= 1.85e+143) {
		tmp = i / ((-1.0 + t_0) / (t_1 * (fma(i, t_1, (beta * alpha)) / t_0)));
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha)) ^ 2.0
	t_1 = Float64(Float64(beta + alpha) + i)
	t_2 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 2.45e+83)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_2) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_2 / i));
	elseif (beta <= 1.85e+143)
		tmp = Float64(i / Float64(Float64(-1.0 + t_0) / Float64(t_1 * Float64(fma(i, t_1, Float64(beta * alpha)) / t_0))));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[Power[N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.45e+83], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.85e+143], N[(i / N[(N[(-1.0 + t$95$0), $MachinePrecision] / N[(t$95$1 * N[(N[(i * t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.44999999999999989e83

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 2.44999999999999989e83 < beta < 1.8500000000000001e143

   1. Initial program 17.1%

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

      1. associate-/l/ 0.7%

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

      2. associate-*l* 0.7%

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

      3. times-frac 37.4%

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

   3. Simplified 68.1%

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

      1. associate-*l/ 68.0%

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

   5. Applied egg-rr 68.0%

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

      1. associate-/l* 68.0%

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

      2. +-commutative 68.0%

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

      3. associate-/r/ 67.9%

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

      4. +-commutative 67.9%

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

      5. +-commutative 67.9%

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

   7. Simplified 67.9%

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

   if 1.8500000000000001e143 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in beta around inf 29.1%

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

      1. associate-/l* 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

      1. add-cbrt-cube 28.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.1%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.0%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 37.8%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 37.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 41.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 41.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 41.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 80.9%

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

       3. associate-*l/ 81.0%

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

       4. +-commutative 81.0%

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

   12. Applied egg-rr 81.0%

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

   13. Taylor expanded in beta around 0 62.6%

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

       1. associate-/l* 80.9%

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

   15. Simplified 80.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+143}:\\ \;\;\;\;\frac{i}{\frac{-1 + {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 5: 83.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_3 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_3 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_3}{i}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_1} \cdot \frac{i \cdot t_0}{t_1}}{-1 + t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) i))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (+ (+ beta alpha) (* 2.0 i)))
        (t_3 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 1.2e+83)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_3)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_3 i))
     (if (<= beta 2e+142)
       (/
        (* (/ (fma i t_0 (* beta alpha)) t_1) (/ (* i t_0) t_1))
        (+ -1.0 (* t_2 t_2)))
       (/ (/ i (/ beta (+ alpha i))) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + i;
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = (beta + alpha) + (2.0 * i);
	double t_3 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 1.2e+83) {
		tmp = (0.0625 + ((((beta + alpha) * t_3) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_3 / i);
	} else if (beta <= 2e+142) {
		tmp = ((fma(i, t_0, (beta * alpha)) / t_1) * ((i * t_0) / t_1)) / (-1.0 + (t_2 * t_2));
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + i)
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	t_3 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 1.2e+83)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_3) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_3 / i));
	elseif (beta <= 2e+142)
		tmp = Float64(Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / t_1) * Float64(Float64(i * t_0) / t_1)) / Float64(-1.0 + Float64(t_2 * t_2)));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.2e+83], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2e+142], N[(N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(i * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.19999999999999996e83

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 1.19999999999999996e83 < beta < 2.0000000000000001e142

   1. Initial program 17.1%

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

      1. times-frac 67.9%

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

      2. +-commutative 67.9%

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

      3. +-commutative 67.9%

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

      4. *-commutative 67.9%

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

      5. fma-udef 67.9%

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

      6. +-commutative 67.9%

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

      7. associate-+l+ 67.9%

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

      8. +-commutative 67.9%

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

      9. *-commutative 67.9%

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

      10. +-commutative 67.9%

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

      11. associate-+l+ 67.9%

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

      12. +-commutative 67.9%

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

      13. fma-udef 67.9%

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

      14. +-commutative 67.9%

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

      15. *-commutative 67.9%

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

      16. fma-udef 67.9%

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

   3. Applied egg-rr 67.9%

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

   if 2.0000000000000001e142 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in beta around inf 29.1%

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

      1. associate-/l* 30.3%

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

      2. unpow2 30.3%

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

   6. Simplified 30.3%

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

      1. add-cbrt-cube 28.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.1%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.0%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 37.8%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 37.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 37.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 41.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 41.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 41.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 80.9%

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

       3. associate-*l/ 81.0%

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

       4. +-commutative 81.0%

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

   12. Applied egg-rr 81.0%

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

   13. Taylor expanded in beta around 0 62.6%

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

       1. associate-/l* 80.9%

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

   15. Simplified 80.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+83}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i \cdot \left(\left(\beta + \alpha\right) + i\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 6: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+80}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_1 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_1}{i}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{i}{-1 + {t_0}^{2}} \cdot \left(\frac{\left(\beta + \alpha\right) + i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i \cdot \left(\beta + i\right)}{t_0}\right)\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+164}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 9.8e+80)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_1)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_1 i))
     (if (<= beta 4.8e+153)
       (*
        (/ i (+ -1.0 (pow t_0 2.0)))
        (*
         (/ (+ (+ beta alpha) i) (+ alpha (fma i 2.0 beta)))
         (/ (* i (+ beta i)) t_0)))
       (if (<= beta 1.2e+164)
         (- (+ 0.0625 (* 0.125 (/ beta i))) (* 0.125 (/ (+ beta alpha) i)))
         (/ (/ i (/ beta (+ alpha i))) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 9.8e+80) {
		tmp = (0.0625 + ((((beta + alpha) * t_1) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_1 / i);
	} else if (beta <= 4.8e+153) {
		tmp = (i / (-1.0 + pow(t_0, 2.0))) * ((((beta + alpha) + i) / (alpha + fma(i, 2.0, beta))) * ((i * (beta + i)) / t_0));
	} else if (beta <= 1.2e+164) {
		tmp = (0.0625 + (0.125 * (beta / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 9.8e+80)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_1) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_1 / i));
	elseif (beta <= 4.8e+153)
		tmp = Float64(Float64(i / Float64(-1.0 + (t_0 ^ 2.0))) * Float64(Float64(Float64(Float64(beta + alpha) + i) / Float64(alpha + fma(i, 2.0, beta))) * Float64(Float64(i * Float64(beta + i)) / t_0)));
	elseif (beta <= 1.2e+164)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	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[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.8e+80], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.8e+153], N[(N[(i / N[(-1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.2e+164], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 9.79999999999999919e80

   1. Initial program 19.4%

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

   2. Taylor expanded in i around inf 40.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 40.7%

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

      2. unpow2 40.7%

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

      3. distribute-lft-out-- 40.7%

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

      4. distribute-lft-out 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in i around -inf 79.8%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   if 9.79999999999999919e80 < beta < 4.79999999999999985e153

   1. Initial program 15.0%

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

      1. associate-/l/ 0.6%

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

      2. associate-*l* 0.6%

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

      3. times-frac 32.6%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in alpha around 0 67.9%

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

   5. Taylor expanded in alpha around 0 67.9%

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

   if 4.79999999999999985e153 < beta < 1.20000000000000005e164

   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 1.9%

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

   4. Taylor expanded in i around inf 62.7%

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

   5. Taylor expanded in alpha around 0 62.3%

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

   6. Taylor expanded in i around 0 62.3%

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

   if 1.20000000000000005e164 < 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 9.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 30.7%

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

      2. unpow2 30.7%

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

   6. Simplified 30.7%

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

      1. add-cbrt-cube 30.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 42.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 42.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 47.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 47.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 47.4%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 89.7%

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

       3. associate-*l/ 89.8%

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

       4. +-commutative 89.8%

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

   12. Applied egg-rr 89.8%

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

   13. Taylor expanded in beta around 0 67.6%

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

       1. associate-/l* 89.8%

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

   15. Simplified 89.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+80}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{i}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(\frac{\left(\beta + \alpha\right) + i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+164}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 7: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := -0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+115}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot t_0 - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{t_0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* -0.0625 (- (+ beta alpha) (+ beta alpha)))))
   (if (<= beta 6.4e+115)
     (+
      (+
       0.0625
       (/
        (-
         (* (+ beta alpha) t_0)
         (* (+ (pow (+ beta alpha) 2.0) -1.0) 0.015625))
        (* i i)))
      (/ t_0 i))
     (/ (/ i (/ beta (+ alpha i))) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 6.4e+115) {
		tmp = (0.0625 + ((((beta + alpha) * t_0) - ((pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_0 / i);
	} else {
		tmp = (i / (beta / (alpha + 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) :: tmp
    t_0 = (-0.0625d0) * ((beta + alpha) - (beta + alpha))
    if (beta <= 6.4d+115) then
        tmp = (0.0625d0 + ((((beta + alpha) * t_0) - ((((beta + alpha) ** 2.0d0) + (-1.0d0)) * 0.015625d0)) / (i * i))) + (t_0 / i)
    else
        tmp = (i / (beta / (alpha + 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 = -0.0625 * ((beta + alpha) - (beta + alpha));
	double tmp;
	if (beta <= 6.4e+115) {
		tmp = (0.0625 + ((((beta + alpha) * t_0) - ((Math.pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_0 / i);
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = -0.0625 * ((beta + alpha) - (beta + alpha))
	tmp = 0
	if beta <= 6.4e+115:
		tmp = (0.0625 + ((((beta + alpha) * t_0) - ((math.pow((beta + alpha), 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_0 / i)
	else:
		tmp = (i / (beta / (alpha + i))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(-0.0625 * Float64(Float64(beta + alpha) - Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 6.4e+115)
		tmp = Float64(Float64(0.0625 + Float64(Float64(Float64(Float64(beta + alpha) * t_0) - Float64(Float64((Float64(beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / Float64(i * i))) + Float64(t_0 / i));
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + i))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = -0.0625 * ((beta + alpha) - (beta + alpha));
	tmp = 0.0;
	if (beta <= 6.4e+115)
		tmp = (0.0625 + ((((beta + alpha) * t_0) - ((((beta + alpha) ^ 2.0) + -1.0) * 0.015625)) / (i * i))) + (t_0 / i);
	else
		tmp = (i / (beta / (alpha + 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[(-0.0625 * N[(N[(beta + alpha), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.4e+115], N[(N[(0.0625 + N[(N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4e115

   1. Initial program 19.5%

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

   2. Taylor expanded in i around inf 39.8%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   3. Step-by-step derivation

      1. fma-def 39.8%

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

      2. unpow2 39.8%

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

      3. distribute-lft-out-- 39.8%

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

      4. distribute-lft-out 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in i around -inf 77.7%

      \[\leadsto \color{blue}{0.0625 + \left(-1 \cdot \frac{-1 \cdot \left(\left(\alpha + \beta\right) \cdot \left(-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)\right) + 0.015625 \cdot \left({\left(\alpha + \beta\right)}^{2} - 1\right)}{{i}^{2}} + -1 \cdot \frac{-0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

   7. Simplified 77.7%

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

   if 6.4e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.3%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 39.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 39.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 39.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 77.7%

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

       3. associate-*l/ 77.8%

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

       4. +-commutative 77.8%

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

   12. Applied egg-rr 77.8%

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

   13. Taylor expanded in beta around 0 62.4%

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

       1. associate-/l* 77.8%

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

   15. Simplified 77.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+115}:\\ \;\;\;\;\left(0.0625 + \frac{\left(\beta + \alpha\right) \cdot \left(-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)\right) - \left({\left(\beta + \alpha\right)}^{2} + -1\right) \cdot 0.015625}{i \cdot i}\right) + \frac{-0.0625 \cdot \left(\left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 8: 84.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.5e+115) 0.0625 (* (/ i beta) (/ (+ alpha i) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.5e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + 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 <= 4.5d+115) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + 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 <= 4.5e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.5e+115:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.5e+115)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + 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 <= 4.5e+115)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + 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, 4.5e+115], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.49999999999999963e115

   1. Initial program 19.5%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 25.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 80.2%

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

   if 4.49999999999999963e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. *-un-lft-identity 35.5%

         \[\leadsto \color{blue}{1 \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      2. associate-/l* 55.3%

         \[\leadsto 1 \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \]

   8. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{1 \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 55.3%

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

      2. associate-/r/ 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 9: 84.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+115) 0.0625 (/ (* i (/ (+ alpha i) beta)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((alpha + i) / beta)) / 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 <= 1.55d+115) then
        tmp = 0.0625d0
    else
        tmp = (i * ((alpha + i) / beta)) / 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 <= 1.55e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((alpha + i) / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.55e+115:
		tmp = 0.0625
	else:
		tmp = (i * ((alpha + i) / beta)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+115)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.55e+115)
		tmp = 0.0625;
	else
		tmp = (i * ((alpha + i) / beta)) / 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, 1.55e+115], 0.0625, N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000002e115

   1. Initial program 19.5%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 25.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 80.2%

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

   if 1.55000000000000002e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.3%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 39.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 39.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 39.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 77.7%

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

       3. associate-*l/ 77.8%

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

       4. +-commutative 77.8%

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

   12. Applied egg-rr 77.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \]

Alternative 10: 84.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.2e+115) 0.0625 (/ (/ i (/ beta (+ alpha i))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta / (alpha + 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 <= 2.2d+115) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta / (alpha + 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 <= 2.2e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta / (alpha + i))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.2e+115:
		tmp = 0.0625
	else:
		tmp = (i / (beta / (alpha + i))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.2e+115)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta / Float64(alpha + 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 <= 2.2e+115)
		tmp = 0.0625;
	else
		tmp = (i / (beta / (alpha + 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, 2.2e+115], 0.0625, N[(N[(i / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2e115

   1. Initial program 19.5%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 25.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 80.2%

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

   if 2.2e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.3%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 39.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 39.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 39.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 77.7%

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

       3. associate-*l/ 77.8%

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

       4. +-commutative 77.8%

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

   12. Applied egg-rr 77.8%

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

   13. Taylor expanded in beta around 0 62.4%

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

       1. associate-/l* 77.8%

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

   15. Simplified 77.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{\alpha + i}}}{\beta}\\ \end{array} \]

Alternative 11: 72.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4e+167) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4e+167) {
		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 <= 4d+167) 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 <= 4e+167) {
		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 <= 4e+167:
		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 <= 4e+167)
		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 <= 4e+167)
		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, 4e+167], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000002e167

   1. Initial program 18.5%

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

      1. associate-/l/ 15.2%

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

      2. associate-*l* 15.1%

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

      3. times-frac 24.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 77.0%

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

   if 4.0000000000000002e167 < 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 9.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 30.7%

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

      2. unpow2 30.7%

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

   6. Simplified 30.7%

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

      1. add-cbrt-cube 30.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 30.7%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 42.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 42.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 42.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 47.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 47.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 47.4%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 89.7%

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

       3. associate-*l/ 89.8%

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

       4. +-commutative 89.8%

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

   12. Applied egg-rr 89.8%

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

   13. Taylor expanded in i around 0 30.5%

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

       1. unpow2 30.5%

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

       2. times-frac 35.8%

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

   15. Simplified 35.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 12: 82.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8e+115) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+115) {
		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 <= 8d+115) 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 <= 8e+115) {
		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 <= 8e+115:
		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 <= 8e+115)
		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 <= 8e+115)
		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, 8e+115], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.0000000000000001e115

   1. Initial program 19.5%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 25.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 80.2%

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

   if 8.0000000000000001e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.3%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 39.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 39.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 39.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 77.7%

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

       3. associate-*l/ 77.8%

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

       4. +-commutative 77.8%

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

   12. Applied egg-rr 77.8%

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

   13. Taylor expanded in i around inf 34.6%

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

       1. unpow2 34.6%

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

       2. unpow2 34.6%

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

       3. times-frac 74.7%

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

   15. Simplified 74.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 13: 82.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.5e+115) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / 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 <= 8.5d+115) then
        tmp = 0.0625d0
    else
        tmp = (i * (i / beta)) / 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 <= 8.5e+115) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+115:
		tmp = 0.0625
	else:
		tmp = (i * (i / beta)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+115)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+115)
		tmp = 0.0625;
	else
		tmp = (i * (i / beta)) / 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, 8.5e+115], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.50000000000000057e115

   1. Initial program 19.5%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 25.2%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in i around inf 80.2%

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

   if 8.50000000000000057e115 < beta

   1. Initial program 2.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 3.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in beta around inf 34.5%

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

      1. associate-/l* 35.5%

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

      2. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}}} \]

      2. associate-/l* 28.4%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}} \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      3. associate-/l* 28.3%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}\right) \cdot \frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

      4. associate-/l* 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\color{blue}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right) \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}}} \]
   9. Step-by-step derivation

      1. associate-*l* 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)}} \]

      2. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \left(\frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      3. associate-/r/ 36.5%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)} \cdot \frac{i}{\frac{\beta}{\frac{\alpha + i}{\beta}}}\right)} \]

      4. associate-/r/ 39.9%

         \[\leadsto \sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \color{blue}{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}\right)} \]

   10. Simplified 39.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right) \cdot \left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)\right)}} \]
   11. Step-by-step derivation

       1. cube-unmult 39.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\right)}^{3}}} \]

       2. rem-cbrt-cube 77.7%

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

       3. associate-*l/ 77.8%

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

       4. +-commutative 77.8%

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

   12. Applied egg-rr 77.8%

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

   13. Taylor expanded in i around inf 74.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 70.6% 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 15.9%

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

   1. associate-/l/ 13.0%

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

   2. associate-*l* 12.9%

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

   3. times-frac 20.7%

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

3. Simplified 41.2%

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

4. Taylor expanded in i around inf 66.9%

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

5. Final simplification 66.9%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023293 
(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)))