Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 84.5%

Time: 16.7s

Alternatives: 5

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: 16.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := -16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\\ t_1 := i + \left(\beta + \alpha\right)\\ t_2 := \beta \cdot -0.125 + -0.00390625 \cdot t_0\\ t_3 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_4 := \sqrt[3]{\frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}{t_3}}\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 - \frac{t_2}{i}\right) - \mathsf{fma}\left(0.0625, \frac{t_2}{\frac{i \cdot i}{t_0}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\beta + \alpha\right), 4 \cdot \left(\beta \cdot \beta + \left({\left(\beta + \alpha\right)}^{2} + -1\right)\right)\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_3, t_3, -1\right)} \cdot \left(\left(t_4 \cdot \left(t_4 \cdot t_4\right)\right) \cdot \frac{t_1}{t_3}\right)\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+169}:\\ \;\;\;\;\left(0.0625 + \frac{0.0625}{\frac{i}{\beta}}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* -16.0 (+ beta (+ beta alpha))))
        (t_1 (+ i (+ beta alpha)))
        (t_2 (+ (* beta -0.125) (* -0.00390625 t_0)))
        (t_3 (+ alpha (fma i 2.0 beta)))
        (t_4 (cbrt (/ (fma i t_1 (* beta alpha)) t_3))))
   (if (<= beta 2e+108)
     (-
      (fma 0.0625 (/ (* beta beta) (* i i)) (- 0.0625 (/ t_2 i)))
      (fma
       0.0625
       (/ t_2 (/ (* i i) t_0))
       (*
        0.00390625
        (/
         (fma
          16.0
          (* beta (+ beta alpha))
          (* 4.0 (+ (* beta beta) (+ (pow (+ beta alpha) 2.0) -1.0))))
         (* i i)))))
     (if (<= beta 3.4e+139)
       (* (/ i (fma t_3 t_3 -1.0)) (* (* t_4 (* t_4 t_4)) (/ t_1 t_3)))
       (if (<= beta 4.8e+169)
         (- (+ 0.0625 (/ 0.0625 (/ i beta))) (* 0.0625 (/ (+ beta alpha) i)))
         (/ (* (/ i beta) (+ i alpha)) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = -16.0 * (beta + (beta + alpha));
	double t_1 = i + (beta + alpha);
	double t_2 = (beta * -0.125) + (-0.00390625 * t_0);
	double t_3 = alpha + fma(i, 2.0, beta);
	double t_4 = cbrt((fma(i, t_1, (beta * alpha)) / t_3));
	double tmp;
	if (beta <= 2e+108) {
		tmp = fma(0.0625, ((beta * beta) / (i * i)), (0.0625 - (t_2 / i))) - fma(0.0625, (t_2 / ((i * i) / t_0)), (0.00390625 * (fma(16.0, (beta * (beta + alpha)), (4.0 * ((beta * beta) + (pow((beta + alpha), 2.0) + -1.0)))) / (i * i))));
	} else if (beta <= 3.4e+139) {
		tmp = (i / fma(t_3, t_3, -1.0)) * ((t_4 * (t_4 * t_4)) * (t_1 / t_3));
	} else if (beta <= 4.8e+169) {
		tmp = (0.0625 + (0.0625 / (i / beta))) - (0.0625 * ((beta + alpha) / i));
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(-16.0 * Float64(beta + Float64(beta + alpha)))
	t_1 = Float64(i + Float64(beta + alpha))
	t_2 = Float64(Float64(beta * -0.125) + Float64(-0.00390625 * t_0))
	t_3 = Float64(alpha + fma(i, 2.0, beta))
	t_4 = cbrt(Float64(fma(i, t_1, Float64(beta * alpha)) / t_3))
	tmp = 0.0
	if (beta <= 2e+108)
		tmp = Float64(fma(0.0625, Float64(Float64(beta * beta) / Float64(i * i)), Float64(0.0625 - Float64(t_2 / i))) - fma(0.0625, Float64(t_2 / Float64(Float64(i * i) / t_0)), Float64(0.00390625 * Float64(fma(16.0, Float64(beta * Float64(beta + alpha)), Float64(4.0 * Float64(Float64(beta * beta) + Float64((Float64(beta + alpha) ^ 2.0) + -1.0)))) / Float64(i * i)))));
	elseif (beta <= 3.4e+139)
		tmp = Float64(Float64(i / fma(t_3, t_3, -1.0)) * Float64(Float64(t_4 * Float64(t_4 * t_4)) * Float64(t_1 / t_3)));
	elseif (beta <= 4.8e+169)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 / Float64(i / beta))) - Float64(0.0625 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(i + alpha)) / 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[(-16.0 * N[(beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta * -0.125), $MachinePrecision] + N[(-0.00390625 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[(i * t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[beta, 2e+108], N[(N[(0.0625 * N[(N[(beta * beta), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 - N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(t$95$2 / N[(N[(i * i), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.00390625 * N[(N[(16.0 * N[(beta * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.4e+139], N[(N[(i / N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.8e+169], N[(N[(0.0625 + N[(0.0625 / N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.0000000000000001e108

   1. Initial program 20.5%

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

   2. Taylor expanded in alpha around 0 21.9%

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

      1. associate-/l* 42.9%

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

      2. unpow2 42.9%

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

      3. +-commutative 42.9%

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

      4. *-commutative 42.9%

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

      5. fma-udef 42.9%

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

   4. Simplified 42.9%

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

   5. Taylor expanded in i around -inf 75.3%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} + \left(0.0625 + -1 \cdot \frac{-0.125 \cdot \beta - 0.00390625 \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)}{i}\right)\right) - \left(0.0625 \cdot \frac{\left(-0.125 \cdot \beta - 0.00390625 \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)\right) \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)}{{i}^{2}} + 0.00390625 \cdot \frac{16 \cdot \left(\beta \cdot \left(\beta + \alpha\right)\right) + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 4 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

   7. Simplified 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 + \left(-\frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)\right)}{i}\right)\right) - \mathsf{fma}\left(0.0625, \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)\right)}{\frac{i \cdot i}{-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\alpha + \beta\right), 4 \cdot \left(\left({\left(\alpha + \beta\right)}^{2} + -1\right) + \beta \cdot \beta\right)\right)}{i \cdot i}\right)} \]

   if 2.0000000000000001e108 < beta < 3.4000000000000002e139

   1. Initial program 22.9%

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

      1. associate-/l/ 1.2%

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

      2. associate-*l* 1.2%

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

      3. times-frac 33.8%

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

   3. Simplified 55.2%

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

      1. add-cube-cbrt 54.9%

         \[\leadsto \frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right) \]

   5. Applied egg-rr 54.9%

      \[\leadsto \frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right) \]

   if 3.4000000000000002e139 < beta < 4.7999999999999997e169

   1. Initial program 0.7%

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

   2. Taylor expanded in i around inf 12.7%

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

      1. +-commutative 12.7%

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

      2. +-commutative 12.7%

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

      3. *-commutative 12.7%

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

      4. fma-def 12.7%

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

      5. distribute-lft-out-- 12.7%

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

      6. +-commutative 12.7%

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

      7. distribute-lft-out 12.7%

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

      8. *-commutative 12.7%

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

      9. unpow2 12.7%

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

   4. Simplified 12.7%

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

   5. Taylor expanded in i around inf 61.3%

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

   6. Taylor expanded in beta around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. associate-/l* 61.0%

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

   8. Simplified 61.0%

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

   if 4.7999999999999997e169 < 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 21.9%

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

   4. Taylor expanded in beta around inf 39.2%

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

      1. associate-/l* 41.2%

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

      2. unpow2 41.2%

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

      3. +-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in beta around 0 39.2%

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

      1. *-commutative 39.2%

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

      2. +-commutative 39.2%

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

      3. associate-*l/ 41.2%

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

      4. unpow2 41.2%

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

      5. +-commutative 41.2%

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

      6. *-commutative 41.2%

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

      7. +-commutative 41.2%

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

      8. associate-/r* 65.2%

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

   9. Simplified 65.2%

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

       1. associate-*r/ 86.7%

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

       2. +-commutative 86.7%

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

   11. Applied egg-rr 86.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 - \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)}{i}\right) - \mathsf{fma}\left(0.0625, \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)}{\frac{i \cdot i}{-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\beta + \alpha\right), 4 \cdot \left(\beta \cdot \beta + \left({\left(\beta + \alpha\right)}^{2} + -1\right)\right)\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+169}:\\ \;\;\;\;\left(0.0625 + \frac{0.0625}{\frac{i}{\beta}}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := -16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\\ t_2 := \beta \cdot -0.125 + -0.00390625 \cdot t_1\\ t_3 := i + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 - \frac{t_2}{i}\right) - \mathsf{fma}\left(0.0625, \frac{t_2}{\frac{i \cdot i}{t_1}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\beta + \alpha\right), 4 \cdot \left(\beta \cdot \beta + \left({\left(\beta + \alpha\right)}^{2} + -1\right)\right)\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_0, t_0, -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, t_3, \beta \cdot \alpha\right)}{t_0} \cdot \frac{t_3}{t_0}\right)\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+169}:\\ \;\;\;\;\left(0.0625 + \frac{0.0625}{\frac{i}{\beta}}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

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 (* -16.0 (+ beta (+ beta alpha))))
        (t_2 (+ (* beta -0.125) (* -0.00390625 t_1)))
        (t_3 (+ i (+ beta alpha))))
   (if (<= beta 8.8e+105)
     (-
      (fma 0.0625 (/ (* beta beta) (* i i)) (- 0.0625 (/ t_2 i)))
      (fma
       0.0625
       (/ t_2 (/ (* i i) t_1))
       (*
        0.00390625
        (/
         (fma
          16.0
          (* beta (+ beta alpha))
          (* 4.0 (+ (* beta beta) (+ (pow (+ beta alpha) 2.0) -1.0))))
         (* i i)))))
     (if (<= beta 3.4e+139)
       (*
        (/ i (fma t_0 t_0 -1.0))
        (* (/ (fma i t_3 (* beta alpha)) t_0) (/ t_3 t_0)))
       (if (<= beta 4.5e+169)
         (- (+ 0.0625 (/ 0.0625 (/ i beta))) (* 0.0625 (/ (+ beta alpha) i)))
         (/ (* (/ i beta) (+ i alpha)) 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 = -16.0 * (beta + (beta + alpha));
	double t_2 = (beta * -0.125) + (-0.00390625 * t_1);
	double t_3 = i + (beta + alpha);
	double tmp;
	if (beta <= 8.8e+105) {
		tmp = fma(0.0625, ((beta * beta) / (i * i)), (0.0625 - (t_2 / i))) - fma(0.0625, (t_2 / ((i * i) / t_1)), (0.00390625 * (fma(16.0, (beta * (beta + alpha)), (4.0 * ((beta * beta) + (pow((beta + alpha), 2.0) + -1.0)))) / (i * i))));
	} else if (beta <= 3.4e+139) {
		tmp = (i / fma(t_0, t_0, -1.0)) * ((fma(i, t_3, (beta * alpha)) / t_0) * (t_3 / t_0));
	} else if (beta <= 4.5e+169) {
		tmp = (0.0625 + (0.0625 / (i / beta))) - (0.0625 * ((beta + alpha) / i));
	} else {
		tmp = ((i / beta) * (i + alpha)) / 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(-16.0 * Float64(beta + Float64(beta + alpha)))
	t_2 = Float64(Float64(beta * -0.125) + Float64(-0.00390625 * t_1))
	t_3 = Float64(i + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 8.8e+105)
		tmp = Float64(fma(0.0625, Float64(Float64(beta * beta) / Float64(i * i)), Float64(0.0625 - Float64(t_2 / i))) - fma(0.0625, Float64(t_2 / Float64(Float64(i * i) / t_1)), Float64(0.00390625 * Float64(fma(16.0, Float64(beta * Float64(beta + alpha)), Float64(4.0 * Float64(Float64(beta * beta) + Float64((Float64(beta + alpha) ^ 2.0) + -1.0)))) / Float64(i * i)))));
	elseif (beta <= 3.4e+139)
		tmp = Float64(Float64(i / fma(t_0, t_0, -1.0)) * Float64(Float64(fma(i, t_3, Float64(beta * alpha)) / t_0) * Float64(t_3 / t_0)));
	elseif (beta <= 4.5e+169)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 / Float64(i / beta))) - Float64(0.0625 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(i + alpha)) / 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[(-16.0 * N[(beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta * -0.125), $MachinePrecision] + N[(-0.00390625 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.8e+105], N[(N[(0.0625 * N[(N[(beta * beta), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 - N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(t$95$2 / N[(N[(i * i), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.00390625 * N[(N[(16.0 * N[(beta * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + N[(N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.4e+139], N[(N[(i / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$3 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.5e+169], N[(N[(0.0625 + N[(0.0625 / N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 8.80000000000000027e105

   1. Initial program 20.5%

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

   2. Taylor expanded in alpha around 0 21.9%

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

      1. associate-/l* 42.9%

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

      2. unpow2 42.9%

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

      3. +-commutative 42.9%

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

      4. *-commutative 42.9%

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

      5. fma-udef 42.9%

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

   4. Simplified 42.9%

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

   5. Taylor expanded in i around -inf 75.3%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} + \left(0.0625 + -1 \cdot \frac{-0.125 \cdot \beta - 0.00390625 \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)}{i}\right)\right) - \left(0.0625 \cdot \frac{\left(-0.125 \cdot \beta - 0.00390625 \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)\right) \cdot \left(-16 \cdot \beta + -16 \cdot \left(\beta + \alpha\right)\right)}{{i}^{2}} + 0.00390625 \cdot \frac{16 \cdot \left(\beta \cdot \left(\beta + \alpha\right)\right) + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 4 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

   7. Simplified 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 + \left(-\frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)\right)}{i}\right)\right) - \mathsf{fma}\left(0.0625, \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)\right)}{\frac{i \cdot i}{-16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\alpha + \beta\right), 4 \cdot \left(\left({\left(\alpha + \beta\right)}^{2} + -1\right) + \beta \cdot \beta\right)\right)}{i \cdot i}\right)} \]

   if 8.80000000000000027e105 < beta < 3.4000000000000002e139

   1. Initial program 22.9%

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

      1. associate-/l/ 1.2%

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

      2. associate-*l* 1.2%

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

      3. times-frac 33.8%

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

   3. Simplified 55.2%

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

   if 3.4000000000000002e139 < beta < 4.5e169

   1. Initial program 0.7%

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

   2. Taylor expanded in i around inf 12.7%

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

      1. +-commutative 12.7%

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

      2. +-commutative 12.7%

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

      3. *-commutative 12.7%

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

      4. fma-def 12.7%

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

      5. distribute-lft-out-- 12.7%

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

      6. +-commutative 12.7%

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

      7. distribute-lft-out 12.7%

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

      8. *-commutative 12.7%

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

      9. unpow2 12.7%

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

   4. Simplified 12.7%

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

   5. Taylor expanded in i around inf 61.3%

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

   6. Taylor expanded in beta around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. associate-/l* 61.0%

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

   8. Simplified 61.0%

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

   if 4.5e169 < 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 21.9%

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

   4. Taylor expanded in beta around inf 39.2%

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

      1. associate-/l* 41.2%

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

      2. unpow2 41.2%

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

      3. +-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in beta around 0 39.2%

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

      1. *-commutative 39.2%

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

      2. +-commutative 39.2%

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

      3. associate-*l/ 41.2%

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

      4. unpow2 41.2%

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

      5. +-commutative 41.2%

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

      6. *-commutative 41.2%

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

      7. +-commutative 41.2%

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

      8. associate-/r* 65.2%

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

   9. Simplified 65.2%

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

       1. associate-*r/ 86.7%

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

       2. +-commutative 86.7%

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

   11. Applied egg-rr 86.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625 - \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)}{i}\right) - \mathsf{fma}\left(0.0625, \frac{\beta \cdot -0.125 + -0.00390625 \cdot \left(-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)}{\frac{i \cdot i}{-16 \cdot \left(\beta + \left(\beta + \alpha\right)\right)}}, 0.00390625 \cdot \frac{\mathsf{fma}\left(16, \beta \cdot \left(\beta + \alpha\right), 4 \cdot \left(\beta \cdot \beta + \left({\left(\beta + \alpha\right)}^{2} + -1\right)\right)\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+169}:\\ \;\;\;\;\left(0.0625 + \frac{0.0625}{\frac{i}{\beta}}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \]

Alternative 3: 85.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+169}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7e+169) 0.0625 (/ (* (/ i beta) (+ i alpha)) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+169) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + 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 <= 7d+169) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * (i + 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 <= 7e+169) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7e+169:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * (i + alpha)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7e+169)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(i + 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 <= 7e+169)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * (i + 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, 7e+169], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.00000000000000038e169

   1. Initial program 19.7%

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

      1. associate-/l/ 17.3%

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

      2. associate-*l* 17.3%

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

      3. times-frac 28.9%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in i around inf 78.5%

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

   if 7.00000000000000038e169 < 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 21.9%

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

   4. Taylor expanded in beta around inf 39.2%

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

      1. associate-/l* 41.2%

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

      2. unpow2 41.2%

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

      3. +-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in beta around 0 39.2%

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

      1. *-commutative 39.2%

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

      2. +-commutative 39.2%

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

      3. associate-*l/ 41.2%

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

      4. unpow2 41.2%

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

      5. +-commutative 41.2%

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

      6. *-commutative 41.2%

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

      7. +-commutative 41.2%

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

      8. associate-/r* 65.2%

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

   9. Simplified 65.2%

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

       1. associate-*r/ 86.7%

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

       2. +-commutative 86.7%

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

   11. Applied egg-rr 86.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+169}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \]

Alternative 4: 83.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+170}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.2e+170) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.2e+170) {
		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 <= 8.2d+170) 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 <= 8.2e+170) {
		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 <= 8.2e+170:
		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 <= 8.2e+170)
		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 <= 8.2e+170)
		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, 8.2e+170], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2000000000000001e170

   1. Initial program 19.7%

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

      1. associate-/l/ 17.3%

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

      2. associate-*l* 17.3%

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

      3. times-frac 28.9%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in i around inf 78.5%

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

   if 8.2000000000000001e170 < 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 21.9%

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

   4. Taylor expanded in beta around inf 39.2%

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

      1. associate-/l* 41.2%

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

      2. unpow2 41.2%

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

      3. +-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in i around inf 39.3%

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

      1. unpow2 39.3%

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

      2. unpow2 39.3%

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

   9. Simplified 39.3%

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

   10. Taylor expanded in i around 0 39.3%

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

       1. unpow2 39.3%

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

       2. unpow2 39.3%

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

       3. times-frac 79.1%

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

   12. Simplified 79.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+170}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5: 71.5% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. associate-/l/ 14.9%

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

   2. associate-*l* 14.8%

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

   3. times-frac 24.8%

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

3. Simplified 40.5%

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

4. Taylor expanded in i around inf 69.2%

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

5. Final simplification 69.2%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023188 
(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)))