Octave 3.8, jcobi/4

Percentage Accurate: 15.5% → 82.7%

Time: 23.6s

Alternatives: 8

Speedup: 53.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 15.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\\ t_2 := \beta \cdot \left(\beta + \alpha\right)\\ t_3 := \beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\\ t_4 := 0.00390625 \cdot t\_3\\ t_5 := \beta + \left(\alpha + i\right)\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+112}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + t\_2 \cdot 16\right)\right) - 0.0625 \cdot \left(t\_3 \cdot \left(t\_4 + t\_1\right)\right)\right) - 0.0625 \cdot t\_2}{i} - t\_1\right) - t\_4}{i}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+149}:\\ \;\;\;\;\frac{i \cdot t\_5}{\mathsf{fma}\left(t\_0, t\_0, -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_5, \beta \cdot \alpha\right)}{t\_0}}{t\_0}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha)))
        (t_1 (* 0.0625 (+ beta (+ beta alpha))))
        (t_2 (* beta (+ beta alpha)))
        (t_3 (+ (* beta -16.0) (* (+ beta alpha) -16.0)))
        (t_4 (* 0.00390625 t_3))
        (t_5 (+ beta (+ alpha i))))
   (if (<= beta 2.3e+112)
     (-
      0.0625
      (/
       (-
        (-
         (/
          (-
           (-
            (*
             0.00390625
             (+
              (* 4.0 (+ -1.0 (pow beta 2.0)))
              (+ (* 4.0 (pow (+ beta alpha) 2.0)) (* t_2 16.0))))
            (* 0.0625 (* t_3 (+ t_4 t_1))))
           (* 0.0625 t_2))
          i)
         t_1)
        t_4)
       i))
     (if (<= beta 3e+149)
       (*
        (/ (* i t_5) (fma t_0 t_0 -1.0))
        (/ (/ (fma i t_5 (* beta alpha)) t_0) t_0))
       (if (<= beta 3e+196)
         (/ (- (+ (* 0.0625 i) (* beta 0.125)) (* (+ beta alpha) 0.125)) i)
         (pow (/ i beta) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = 0.0625 * (beta + (beta + alpha));
	double t_2 = beta * (beta + alpha);
	double t_3 = (beta * -16.0) + ((beta + alpha) * -16.0);
	double t_4 = 0.00390625 * t_3;
	double t_5 = beta + (alpha + i);
	double tmp;
	if (beta <= 2.3e+112) {
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + pow(beta, 2.0))) + ((4.0 * pow((beta + alpha), 2.0)) + (t_2 * 16.0)))) - (0.0625 * (t_3 * (t_4 + t_1)))) - (0.0625 * t_2)) / i) - t_1) - t_4) / i);
	} else if (beta <= 3e+149) {
		tmp = ((i * t_5) / fma(t_0, t_0, -1.0)) * ((fma(i, t_5, (beta * alpha)) / t_0) / t_0);
	} else if (beta <= 3e+196) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(0.0625 * Float64(beta + Float64(beta + alpha)))
	t_2 = Float64(beta * Float64(beta + alpha))
	t_3 = Float64(Float64(beta * -16.0) + Float64(Float64(beta + alpha) * -16.0))
	t_4 = Float64(0.00390625 * t_3)
	t_5 = Float64(beta + Float64(alpha + i))
	tmp = 0.0
	if (beta <= 2.3e+112)
		tmp = Float64(0.0625 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.00390625 * Float64(Float64(4.0 * Float64(-1.0 + (beta ^ 2.0))) + Float64(Float64(4.0 * (Float64(beta + alpha) ^ 2.0)) + Float64(t_2 * 16.0)))) - Float64(0.0625 * Float64(t_3 * Float64(t_4 + t_1)))) - Float64(0.0625 * t_2)) / i) - t_1) - t_4) / i));
	elseif (beta <= 3e+149)
		tmp = Float64(Float64(Float64(i * t_5) / fma(t_0, t_0, -1.0)) * Float64(Float64(fma(i, t_5, Float64(beta * alpha)) / t_0) / t_0));
	elseif (beta <= 3e+196)
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(beta * 0.125)) - Float64(Float64(beta + alpha) * 0.125)) / i);
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0625 * N[(beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta * -16.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.00390625 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(beta + N[(alpha + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.3e+112], N[(0.0625 - N[(N[(N[(N[(N[(N[(N[(0.00390625 * N[(N[(4.0 * N[(-1.0 + N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(t$95$3 * N[(t$95$4 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * t$95$2), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - t$95$1), $MachinePrecision] - t$95$4), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+149], N[(N[(N[(i * t$95$5), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$5 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3e+196], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(beta * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.3e112

   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/ 16.1%

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

      2. associate-*l* 16.1%

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

      3. associate-/l* 16.2%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in alpha around 0 45.8%

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

   6. Taylor expanded in i around -inf 78.9%

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

   if 2.3e112 < beta < 3.00000000000000003e149

   1. Initial program 0.8%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 32.5%

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

   3. Simplified 32.8%

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

   if 3.00000000000000003e149 < beta < 2.9999999999999999e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in i around inf 55.4%

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

   6. Taylor expanded in i around 0 55.4%

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

   7. Taylor expanded in alpha around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

   if 2.9999999999999999e196 < 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. associate-/l* 0.0%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in alpha around 0 26.1%

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

   6. Taylor expanded in beta around -inf 34.9%

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

   7. Taylor expanded in beta around inf 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. times-frac 79.2%

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

      4. unpow2 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+112}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(\beta \cdot \left(\beta + \alpha\right)\right) \cdot 16\right)\right) - 0.0625 \cdot \left(\left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) \cdot \left(0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) + 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)\right)\right) - 0.0625 \cdot \left(\beta \cdot \left(\beta + \alpha\right)\right)}{i} - 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right) - 0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right)}{i}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+149}:\\ \;\;\;\;\frac{i \cdot \left(\beta + \left(\alpha + i\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(\alpha + i\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\\ t_1 := \beta \cdot \left(\beta + \alpha\right)\\ t_2 := \beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\\ t_3 := 0.00390625 \cdot t\_2\\ t_4 := {\left(\beta + 2 \cdot i\right)}^{2}\\ \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+116}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + t\_1 \cdot 16\right)\right) - 0.0625 \cdot \left(t\_2 \cdot \left(t\_3 + t\_0\right)\right)\right) - 0.0625 \cdot t\_1}{i} - t\_0\right) - t\_3}{i}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+149}:\\ \;\;\;\;i \cdot \left(\frac{i}{t\_4} \cdot \frac{{\left(\beta + i\right)}^{2}}{-1 + t\_4}\right)\\ \mathbf{elif}\;\beta \leq 1.12 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.0625 (+ beta (+ beta alpha))))
        (t_1 (* beta (+ beta alpha)))
        (t_2 (+ (* beta -16.0) (* (+ beta alpha) -16.0)))
        (t_3 (* 0.00390625 t_2))
        (t_4 (pow (+ beta (* 2.0 i)) 2.0)))
   (if (<= beta 2.7e+116)
     (-
      0.0625
      (/
       (-
        (-
         (/
          (-
           (-
            (*
             0.00390625
             (+
              (* 4.0 (+ -1.0 (pow beta 2.0)))
              (+ (* 4.0 (pow (+ beta alpha) 2.0)) (* t_1 16.0))))
            (* 0.0625 (* t_2 (+ t_3 t_0))))
           (* 0.0625 t_1))
          i)
         t_0)
        t_3)
       i))
     (if (<= beta 1.5e+149)
       (* i (* (/ i t_4) (/ (pow (+ beta i) 2.0) (+ -1.0 t_4))))
       (if (<= beta 1.12e+197)
         (/ (- (+ (* 0.0625 i) (* beta 0.125)) (* (+ beta alpha) 0.125)) i)
         (pow (/ i beta) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.0625 * (beta + (beta + alpha));
	double t_1 = beta * (beta + alpha);
	double t_2 = (beta * -16.0) + ((beta + alpha) * -16.0);
	double t_3 = 0.00390625 * t_2;
	double t_4 = pow((beta + (2.0 * i)), 2.0);
	double tmp;
	if (beta <= 2.7e+116) {
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + pow(beta, 2.0))) + ((4.0 * pow((beta + alpha), 2.0)) + (t_1 * 16.0)))) - (0.0625 * (t_2 * (t_3 + t_0)))) - (0.0625 * t_1)) / i) - t_0) - t_3) / i);
	} else if (beta <= 1.5e+149) {
		tmp = i * ((i / t_4) * (pow((beta + i), 2.0) / (-1.0 + t_4)));
	} else if (beta <= 1.12e+197) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 0.0625d0 * (beta + (beta + alpha))
    t_1 = beta * (beta + alpha)
    t_2 = (beta * (-16.0d0)) + ((beta + alpha) * (-16.0d0))
    t_3 = 0.00390625d0 * t_2
    t_4 = (beta + (2.0d0 * i)) ** 2.0d0
    if (beta <= 2.7d+116) then
        tmp = 0.0625d0 - (((((((0.00390625d0 * ((4.0d0 * ((-1.0d0) + (beta ** 2.0d0))) + ((4.0d0 * ((beta + alpha) ** 2.0d0)) + (t_1 * 16.0d0)))) - (0.0625d0 * (t_2 * (t_3 + t_0)))) - (0.0625d0 * t_1)) / i) - t_0) - t_3) / i)
    else if (beta <= 1.5d+149) then
        tmp = i * ((i / t_4) * (((beta + i) ** 2.0d0) / ((-1.0d0) + t_4)))
    else if (beta <= 1.12d+197) then
        tmp = (((0.0625d0 * i) + (beta * 0.125d0)) - ((beta + alpha) * 0.125d0)) / i
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.0625 * (beta + (beta + alpha));
	double t_1 = beta * (beta + alpha);
	double t_2 = (beta * -16.0) + ((beta + alpha) * -16.0);
	double t_3 = 0.00390625 * t_2;
	double t_4 = Math.pow((beta + (2.0 * i)), 2.0);
	double tmp;
	if (beta <= 2.7e+116) {
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + Math.pow(beta, 2.0))) + ((4.0 * Math.pow((beta + alpha), 2.0)) + (t_1 * 16.0)))) - (0.0625 * (t_2 * (t_3 + t_0)))) - (0.0625 * t_1)) / i) - t_0) - t_3) / i);
	} else if (beta <= 1.5e+149) {
		tmp = i * ((i / t_4) * (Math.pow((beta + i), 2.0) / (-1.0 + t_4)));
	} else if (beta <= 1.12e+197) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.0625 * (beta + (beta + alpha))
	t_1 = beta * (beta + alpha)
	t_2 = (beta * -16.0) + ((beta + alpha) * -16.0)
	t_3 = 0.00390625 * t_2
	t_4 = math.pow((beta + (2.0 * i)), 2.0)
	tmp = 0
	if beta <= 2.7e+116:
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + math.pow(beta, 2.0))) + ((4.0 * math.pow((beta + alpha), 2.0)) + (t_1 * 16.0)))) - (0.0625 * (t_2 * (t_3 + t_0)))) - (0.0625 * t_1)) / i) - t_0) - t_3) / i)
	elif beta <= 1.5e+149:
		tmp = i * ((i / t_4) * (math.pow((beta + i), 2.0) / (-1.0 + t_4)))
	elif beta <= 1.12e+197:
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.0625 * Float64(beta + Float64(beta + alpha)))
	t_1 = Float64(beta * Float64(beta + alpha))
	t_2 = Float64(Float64(beta * -16.0) + Float64(Float64(beta + alpha) * -16.0))
	t_3 = Float64(0.00390625 * t_2)
	t_4 = Float64(beta + Float64(2.0 * i)) ^ 2.0
	tmp = 0.0
	if (beta <= 2.7e+116)
		tmp = Float64(0.0625 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.00390625 * Float64(Float64(4.0 * Float64(-1.0 + (beta ^ 2.0))) + Float64(Float64(4.0 * (Float64(beta + alpha) ^ 2.0)) + Float64(t_1 * 16.0)))) - Float64(0.0625 * Float64(t_2 * Float64(t_3 + t_0)))) - Float64(0.0625 * t_1)) / i) - t_0) - t_3) / i));
	elseif (beta <= 1.5e+149)
		tmp = Float64(i * Float64(Float64(i / t_4) * Float64((Float64(beta + i) ^ 2.0) / Float64(-1.0 + t_4))));
	elseif (beta <= 1.12e+197)
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(beta * 0.125)) - Float64(Float64(beta + alpha) * 0.125)) / i);
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.0625 * (beta + (beta + alpha));
	t_1 = beta * (beta + alpha);
	t_2 = (beta * -16.0) + ((beta + alpha) * -16.0);
	t_3 = 0.00390625 * t_2;
	t_4 = (beta + (2.0 * i)) ^ 2.0;
	tmp = 0.0;
	if (beta <= 2.7e+116)
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + (beta ^ 2.0))) + ((4.0 * ((beta + alpha) ^ 2.0)) + (t_1 * 16.0)))) - (0.0625 * (t_2 * (t_3 + t_0)))) - (0.0625 * t_1)) / i) - t_0) - t_3) / i);
	elseif (beta <= 1.5e+149)
		tmp = i * ((i / t_4) * (((beta + i) ^ 2.0) / (-1.0 + t_4)));
	elseif (beta <= 1.12e+197)
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.0625 * N[(beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta * -16.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.00390625 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 2.7e+116], N[(0.0625 - N[(N[(N[(N[(N[(N[(N[(0.00390625 * N[(N[(4.0 * N[(-1.0 + N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(t$95$2 * N[(t$95$3 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * t$95$1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+149], N[(i * N[(N[(i / t$95$4), $MachinePrecision] * N[(N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.12e+197], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(beta * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.7e116

   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/ 16.1%

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

      2. associate-*l* 16.1%

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

      3. associate-/l* 16.2%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in alpha around 0 45.8%

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

   6. Taylor expanded in i around -inf 78.9%

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

   if 2.7e116 < beta < 1.50000000000000002e149

   1. Initial program 0.8%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in alpha around 0 0.0%

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

      1. times-frac 35.0%

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

      2. sub-neg 35.0%

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

      3. metadata-eval 35.0%

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

   7. Simplified 35.0%

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

   if 1.50000000000000002e149 < beta < 1.1200000000000001e197

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in i around inf 55.4%

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

   6. Taylor expanded in i around 0 55.4%

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

   7. Taylor expanded in alpha around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

   if 1.1200000000000001e197 < 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. associate-/l* 0.0%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in alpha around 0 26.1%

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

   6. Taylor expanded in beta around -inf 34.9%

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

   7. Taylor expanded in beta around inf 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. times-frac 79.2%

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

      4. unpow2 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+116}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(\beta \cdot \left(\beta + \alpha\right)\right) \cdot 16\right)\right) - 0.0625 \cdot \left(\left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) \cdot \left(0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) + 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)\right)\right) - 0.0625 \cdot \left(\beta \cdot \left(\beta + \alpha\right)\right)}{i} - 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right) - 0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right)}{i}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+149}:\\ \;\;\;\;i \cdot \left(\frac{i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}}\right)\\ \mathbf{elif}\;\beta \leq 1.12 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\\ t_1 := {\left(\beta + 2 \cdot i\right)}^{2}\\ t_2 := 0.00390625 \cdot t\_0\\ t_3 := 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\\ t_4 := \beta \cdot \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+117}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + t\_4 \cdot 16\right)\right) - 0.0625 \cdot \left(t\_0 \cdot \left(t\_2 + t\_3\right)\right)\right) - 0.0625 \cdot t\_4}{i} - t\_3\right) - t\_2}{i}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(\beta + i\right)}{-1 + t\_1} \cdot \frac{\beta + i}{t\_1}\right)\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* beta -16.0) (* (+ beta alpha) -16.0)))
        (t_1 (pow (+ beta (* 2.0 i)) 2.0))
        (t_2 (* 0.00390625 t_0))
        (t_3 (* 0.0625 (+ beta (+ beta alpha))))
        (t_4 (* beta (+ beta alpha))))
   (if (<= beta 1.8e+117)
     (-
      0.0625
      (/
       (-
        (-
         (/
          (-
           (-
            (*
             0.00390625
             (+
              (* 4.0 (+ -1.0 (pow beta 2.0)))
              (+ (* 4.0 (pow (+ beta alpha) 2.0)) (* t_4 16.0))))
            (* 0.0625 (* t_0 (+ t_2 t_3))))
           (* 0.0625 t_4))
          i)
         t_3)
        t_2)
       i))
     (if (<= beta 1.6e+149)
       (* i (* (/ (* i (+ beta i)) (+ -1.0 t_1)) (/ (+ beta i) t_1)))
       (if (<= beta 1.65e+196)
         (/ (- (+ (* 0.0625 i) (* beta 0.125)) (* (+ beta alpha) 0.125)) i)
         (pow (/ i beta) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta * -16.0) + ((beta + alpha) * -16.0);
	double t_1 = pow((beta + (2.0 * i)), 2.0);
	double t_2 = 0.00390625 * t_0;
	double t_3 = 0.0625 * (beta + (beta + alpha));
	double t_4 = beta * (beta + alpha);
	double tmp;
	if (beta <= 1.8e+117) {
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + pow(beta, 2.0))) + ((4.0 * pow((beta + alpha), 2.0)) + (t_4 * 16.0)))) - (0.0625 * (t_0 * (t_2 + t_3)))) - (0.0625 * t_4)) / i) - t_3) - t_2) / i);
	} else if (beta <= 1.6e+149) {
		tmp = i * (((i * (beta + i)) / (-1.0 + t_1)) * ((beta + i) / t_1));
	} else if (beta <= 1.65e+196) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (beta * (-16.0d0)) + ((beta + alpha) * (-16.0d0))
    t_1 = (beta + (2.0d0 * i)) ** 2.0d0
    t_2 = 0.00390625d0 * t_0
    t_3 = 0.0625d0 * (beta + (beta + alpha))
    t_4 = beta * (beta + alpha)
    if (beta <= 1.8d+117) then
        tmp = 0.0625d0 - (((((((0.00390625d0 * ((4.0d0 * ((-1.0d0) + (beta ** 2.0d0))) + ((4.0d0 * ((beta + alpha) ** 2.0d0)) + (t_4 * 16.0d0)))) - (0.0625d0 * (t_0 * (t_2 + t_3)))) - (0.0625d0 * t_4)) / i) - t_3) - t_2) / i)
    else if (beta <= 1.6d+149) then
        tmp = i * (((i * (beta + i)) / ((-1.0d0) + t_1)) * ((beta + i) / t_1))
    else if (beta <= 1.65d+196) then
        tmp = (((0.0625d0 * i) + (beta * 0.125d0)) - ((beta + alpha) * 0.125d0)) / i
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta * -16.0) + ((beta + alpha) * -16.0);
	double t_1 = Math.pow((beta + (2.0 * i)), 2.0);
	double t_2 = 0.00390625 * t_0;
	double t_3 = 0.0625 * (beta + (beta + alpha));
	double t_4 = beta * (beta + alpha);
	double tmp;
	if (beta <= 1.8e+117) {
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + Math.pow(beta, 2.0))) + ((4.0 * Math.pow((beta + alpha), 2.0)) + (t_4 * 16.0)))) - (0.0625 * (t_0 * (t_2 + t_3)))) - (0.0625 * t_4)) / i) - t_3) - t_2) / i);
	} else if (beta <= 1.6e+149) {
		tmp = i * (((i * (beta + i)) / (-1.0 + t_1)) * ((beta + i) / t_1));
	} else if (beta <= 1.65e+196) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta * -16.0) + ((beta + alpha) * -16.0)
	t_1 = math.pow((beta + (2.0 * i)), 2.0)
	t_2 = 0.00390625 * t_0
	t_3 = 0.0625 * (beta + (beta + alpha))
	t_4 = beta * (beta + alpha)
	tmp = 0
	if beta <= 1.8e+117:
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + math.pow(beta, 2.0))) + ((4.0 * math.pow((beta + alpha), 2.0)) + (t_4 * 16.0)))) - (0.0625 * (t_0 * (t_2 + t_3)))) - (0.0625 * t_4)) / i) - t_3) - t_2) / i)
	elif beta <= 1.6e+149:
		tmp = i * (((i * (beta + i)) / (-1.0 + t_1)) * ((beta + i) / t_1))
	elif beta <= 1.65e+196:
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta * -16.0) + Float64(Float64(beta + alpha) * -16.0))
	t_1 = Float64(beta + Float64(2.0 * i)) ^ 2.0
	t_2 = Float64(0.00390625 * t_0)
	t_3 = Float64(0.0625 * Float64(beta + Float64(beta + alpha)))
	t_4 = Float64(beta * Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.8e+117)
		tmp = Float64(0.0625 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.00390625 * Float64(Float64(4.0 * Float64(-1.0 + (beta ^ 2.0))) + Float64(Float64(4.0 * (Float64(beta + alpha) ^ 2.0)) + Float64(t_4 * 16.0)))) - Float64(0.0625 * Float64(t_0 * Float64(t_2 + t_3)))) - Float64(0.0625 * t_4)) / i) - t_3) - t_2) / i));
	elseif (beta <= 1.6e+149)
		tmp = Float64(i * Float64(Float64(Float64(i * Float64(beta + i)) / Float64(-1.0 + t_1)) * Float64(Float64(beta + i) / t_1)));
	elseif (beta <= 1.65e+196)
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(beta * 0.125)) - Float64(Float64(beta + alpha) * 0.125)) / i);
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta * -16.0) + ((beta + alpha) * -16.0);
	t_1 = (beta + (2.0 * i)) ^ 2.0;
	t_2 = 0.00390625 * t_0;
	t_3 = 0.0625 * (beta + (beta + alpha));
	t_4 = beta * (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.8e+117)
		tmp = 0.0625 - (((((((0.00390625 * ((4.0 * (-1.0 + (beta ^ 2.0))) + ((4.0 * ((beta + alpha) ^ 2.0)) + (t_4 * 16.0)))) - (0.0625 * (t_0 * (t_2 + t_3)))) - (0.0625 * t_4)) / i) - t_3) - t_2) / i);
	elseif (beta <= 1.6e+149)
		tmp = i * (((i * (beta + i)) / (-1.0 + t_1)) * ((beta + i) / t_1));
	elseif (beta <= 1.65e+196)
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta * -16.0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(0.00390625 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(0.0625 * N[(beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.8e+117], N[(0.0625 - N[(N[(N[(N[(N[(N[(N[(0.00390625 * N[(N[(4.0 * N[(-1.0 + N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * t$95$4), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.6e+149], N[(i * N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.65e+196], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(beta * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.80000000000000006e117

   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/ 16.1%

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

      2. associate-*l* 16.1%

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

      3. associate-/l* 16.2%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in alpha around 0 45.8%

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

   6. Taylor expanded in i around -inf 78.9%

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

   if 1.80000000000000006e117 < beta < 1.6000000000000001e149

   1. Initial program 0.8%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in alpha around 0 65.8%

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

   6. Taylor expanded in alpha around 0 34.2%

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

   if 1.6000000000000001e149 < beta < 1.6500000000000001e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in i around inf 55.4%

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

   6. Taylor expanded in i around 0 55.4%

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

   7. Taylor expanded in alpha around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

   if 1.6500000000000001e196 < 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. associate-/l* 0.0%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in alpha around 0 26.1%

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

   6. Taylor expanded in beta around -inf 34.9%

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

   7. Taylor expanded in beta around inf 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. times-frac 79.2%

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

      4. unpow2 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+117}:\\ \;\;\;\;0.0625 - \frac{\left(\frac{\left(0.00390625 \cdot \left(4 \cdot \left(-1 + {\beta}^{2}\right) + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(\beta \cdot \left(\beta + \alpha\right)\right) \cdot 16\right)\right) - 0.0625 \cdot \left(\left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) \cdot \left(0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right) + 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right)\right)\right) - 0.0625 \cdot \left(\beta \cdot \left(\beta + \alpha\right)\right)}{i} - 0.0625 \cdot \left(\beta + \left(\beta + \alpha\right)\right)\right) - 0.00390625 \cdot \left(\beta \cdot -16 + \left(\beta + \alpha\right) \cdot -16\right)}{i}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(\beta + i\right)}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{\beta + i}{{\left(\beta + 2 \cdot i\right)}^{2}}\right)\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := {\left(\beta + 2 \cdot i\right)}^{2}\\ \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.06 \cdot 10^{+148}:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(\beta + i\right)}{-1 + t\_0} \cdot \frac{\beta + i}{t\_0}\right)\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (pow (+ beta (* 2.0 i)) 2.0)))
   (if (<= beta 3.9e+121)
     0.0625
     (if (<= beta 1.06e+148)
       (* i (* (/ (* i (+ beta i)) (+ -1.0 t_0)) (/ (+ beta i) t_0)))
       (if (<= beta 1.55e+196)
         (/ (- (+ (* 0.0625 i) (* beta 0.125)) (* (+ beta alpha) 0.125)) i)
         (pow (/ i beta) 2.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = pow((beta + (2.0 * i)), 2.0);
	double tmp;
	if (beta <= 3.9e+121) {
		tmp = 0.0625;
	} else if (beta <= 1.06e+148) {
		tmp = i * (((i * (beta + i)) / (-1.0 + t_0)) * ((beta + i) / t_0));
	} else if (beta <= 1.55e+196) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + (2.0d0 * i)) ** 2.0d0
    if (beta <= 3.9d+121) then
        tmp = 0.0625d0
    else if (beta <= 1.06d+148) then
        tmp = i * (((i * (beta + i)) / ((-1.0d0) + t_0)) * ((beta + i) / t_0))
    else if (beta <= 1.55d+196) then
        tmp = (((0.0625d0 * i) + (beta * 0.125d0)) - ((beta + alpha) * 0.125d0)) / i
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = Math.pow((beta + (2.0 * i)), 2.0);
	double tmp;
	if (beta <= 3.9e+121) {
		tmp = 0.0625;
	} else if (beta <= 1.06e+148) {
		tmp = i * (((i * (beta + i)) / (-1.0 + t_0)) * ((beta + i) / t_0));
	} else if (beta <= 1.55e+196) {
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = math.pow((beta + (2.0 * i)), 2.0)
	tmp = 0
	if beta <= 3.9e+121:
		tmp = 0.0625
	elif beta <= 1.06e+148:
		tmp = i * (((i * (beta + i)) / (-1.0 + t_0)) * ((beta + i) / t_0))
	elif beta <= 1.55e+196:
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i)) ^ 2.0
	tmp = 0.0
	if (beta <= 3.9e+121)
		tmp = 0.0625;
	elseif (beta <= 1.06e+148)
		tmp = Float64(i * Float64(Float64(Float64(i * Float64(beta + i)) / Float64(-1.0 + t_0)) * Float64(Float64(beta + i) / t_0)));
	elseif (beta <= 1.55e+196)
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(beta * 0.125)) - Float64(Float64(beta + alpha) * 0.125)) / i);
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + (2.0 * i)) ^ 2.0;
	tmp = 0.0;
	if (beta <= 3.9e+121)
		tmp = 0.0625;
	elseif (beta <= 1.06e+148)
		tmp = i * (((i * (beta + i)) / (-1.0 + t_0)) * ((beta + i) / t_0));
	elseif (beta <= 1.55e+196)
		tmp = (((0.0625 * i) + (beta * 0.125)) - ((beta + alpha) * 0.125)) / i;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[Power[N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 3.9e+121], 0.0625, If[LessEqual[beta, 1.06e+148], N[(i * N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.55e+196], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(beta * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 3.89999999999999984e121

   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/ 16.1%

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

      2. associate-*l* 16.1%

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

      3. associate-/l* 16.2%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in i around inf 83.4%

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

   if 3.89999999999999984e121 < beta < 1.06e148

   1. Initial program 0.8%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in alpha around 0 65.8%

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

   6. Taylor expanded in alpha around 0 34.2%

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

   if 1.06e148 < beta < 1.55000000000000005e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-/l* 0.0%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in i around inf 55.4%

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

   6. Taylor expanded in i around 0 55.4%

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

   7. Taylor expanded in alpha around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

   if 1.55000000000000005e196 < 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. associate-/l* 0.0%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in alpha around 0 26.1%

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

   6. Taylor expanded in beta around -inf 34.9%

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

   7. Taylor expanded in beta around inf 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. times-frac 79.2%

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

      4. unpow2 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.06 \cdot 10^{+148}:\\ \;\;\;\;i \cdot \left(\frac{i \cdot \left(\beta + i\right)}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{\beta + i}{{\left(\beta + 2 \cdot i\right)}^{2}}\right)\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+196) 0.0625 (pow (/ i beta) 2.0)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+196) {
		tmp = 0.0625;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.55d+196) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+196) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.55e+196:
		tmp = 0.0625
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+196)
		tmp = 0.0625;
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.55e+196)
		tmp = 0.0625;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.55e+196], 0.0625, N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000005e196

   1. Initial program 15.5%

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

      1. associate-/l/ 14.7%

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

      2. associate-*l* 14.6%

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

      3. associate-/l* 14.8%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in i around inf 80.6%

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

   if 1.55000000000000005e196 < 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. associate-/l* 0.0%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in alpha around 0 26.1%

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

   6. Taylor expanded in beta around -inf 34.9%

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

   7. Taylor expanded in beta around inf 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. times-frac 79.2%

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

      4. unpow2 79.2%

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

   9. Simplified 79.2%

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

Alternative 6: 77.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((0.0625d0 * i) + (beta * 0.125d0)) - ((beta + alpha) * 0.125d0)) / i
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.6%

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

   1. associate-/l/ 12.9%

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

   2. associate-*l* 12.8%

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

   3. associate-/l* 12.9%

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

3. Simplified 41.1%

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

5. Taylor expanded in i around inf 79.1%

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

6. Taylor expanded in i around 0 79.1%

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

7. Taylor expanded in alpha around 0 74.8%

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

   1. *-commutative 74.8%

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

9. Simplified 74.8%

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

10. Final simplification 74.8%

    \[\leadsto \frac{\left(0.0625 \cdot i + \beta \cdot 0.125\right) - \left(\beta + \alpha\right) \cdot 0.125}{i} \]
11. Add Preprocessing

Alternative 7: 73.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.5e+200) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+200) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.5d+200) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+200) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.5e+200:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+200)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.5e+200)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+200], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e200

   1. Initial program 15.4%

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

      1. associate-/l/ 14.6%

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

      2. associate-*l* 14.6%

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

      3. associate-/l* 14.7%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in i around inf 80.3%

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

   if 5.5e200 < 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. associate-/l* 0.0%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in i around inf 43.2%

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

   6. Taylor expanded in i around 0 43.2%

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

   7. Taylor expanded in i around 0 37.5%

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

      1. fma-neg 37.5%

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

      2. distribute-lft-in 37.5%

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

      3. fma-neg 37.5%

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

      4. associate-*r* 37.5%

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

      5. metadata-eval 37.5%

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

      6. +-inverses 37.5%

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

   9. Simplified 37.5%

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

Alternative 8: 70.2% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.6%

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

   1. associate-/l/ 12.9%

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

   2. associate-*l* 12.8%

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

   3. associate-/l* 12.9%

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

3. Simplified 41.1%

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

5. Taylor expanded in i around inf 71.7%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))