Octave 3.8, jcobi/4

Percentage Accurate: 15.3% → 84.4%

Time: 35.2s

Alternatives: 20

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: 15.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := {\left(\beta + i\right)}^{2}\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := {\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{t_2} \cdot \log \left(e^{\frac{t_0}{-1 + t_2}}\right)\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{t_0}}}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (pow (+ beta i) 2.0))
        (t_1 (+ (+ beta alpha) (* i 2.0)))
        (t_2 (pow (fma i 2.0 beta) 2.0)))
   (if (<= beta 3.7e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (* (/ (* i i) t_2) (log (exp (/ t_0 (+ -1.0 t_2)))))
       (if (<= beta 2.6e+98)
         0.0625
         (if (<= beta 1.75e+141)
           (/
            (/
             (* i i)
             (/ (fma 4.0 (* beta i) (fma 4.0 (* i i) (* beta beta))) t_0))
            (+ -1.0 (* t_1 t_1)))
           (if (<= beta 6.5e+170)
             (-
              (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
              (* 0.125 (/ (+ beta alpha) i)))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = pow((beta + i), 2.0);
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = pow(fma(i, 2.0, beta), 2.0);
	double tmp;
	if (beta <= 3.7e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((i * i) / t_2) * log(exp((t_0 / (-1.0 + t_2))));
	} else if (beta <= 2.6e+98) {
		tmp = 0.0625;
	} else if (beta <= 1.75e+141) {
		tmp = ((i * i) / (fma(4.0, (beta * i), fma(4.0, (i * i), (beta * beta))) / t_0)) / (-1.0 + (t_1 * t_1));
	} else if (beta <= 6.5e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + i) ^ 2.0
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_2 = fma(i, 2.0, beta) ^ 2.0
	tmp = 0.0
	if (beta <= 3.7e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(i * i) / t_2) * log(exp(Float64(t_0 / Float64(-1.0 + t_2)))));
	elseif (beta <= 2.6e+98)
		tmp = 0.0625;
	elseif (beta <= 1.75e+141)
		tmp = Float64(Float64(Float64(i * i) / Float64(fma(4.0, Float64(beta * i), fma(4.0, Float64(i * i), Float64(beta * beta))) / t_0)) / Float64(-1.0 + Float64(t_1 * t_1)));
	elseif (beta <= 6.5e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(i * 2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 3.7e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(i * i), $MachinePrecision] / t$95$2), $MachinePrecision] * N[Log[N[Exp[N[(t$95$0 / N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.6e+98], 0.0625, If[LessEqual[beta, 1.75e+141], N[(N[(N[(i * i), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.5e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if beta < 3.69999999999999968e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 3.69999999999999968e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

      1. associate-/l/ 74.1%

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

      2. associate-*l* 74.1%

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

      3. times-frac 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in alpha around 0 75.1%

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

      1. times-frac 75.3%

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

      2. unpow2 75.3%

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

      3. +-commutative 75.3%

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

      4. *-commutative 75.3%

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

      5. fma-udef 75.3%

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

      6. sub-neg 75.3%

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

      7. +-commutative 75.3%

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

      8. *-commutative 75.3%

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

      9. fma-udef 75.3%

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

      10. metadata-eval 75.3%

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

   6. Simplified 75.3%

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

      1. add-log-exp 75.9%

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

      2. +-commutative 75.9%

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

   8. Applied egg-rr 75.9%

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

   if 3.2999999999999999e69 < beta < 2.6e98

   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. times-frac 0.8%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in i around inf 74.4%

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

   if 2.6e98 < beta < 1.75e141

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around 0 15.9%

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

      1. associate-/l* 84.9%

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

      2. unpow2 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. fma-udef 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in i around 0 85.3%

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

      1. fma-def 85.3%

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

      2. fma-def 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

   7. Simplified 85.3%

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

   if 1.75e141 < beta < 6.5e170

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in i around inf 84.4%

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

   5. Taylor expanded in alpha around 0 76.2%

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

   if 6.5e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \log \left(e^{\frac{{\left(\beta + i\right)}^{2}}{-1 + {\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 2: 83.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+58}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_0, t_0, -1\right)} \cdot \frac{\left(i \cdot {\left(i + \alpha\right)}^{2} + \left(\beta \cdot \beta\right) \cdot \left(i + \alpha\right)\right) + \beta \cdot \left(i \cdot \left(i + \alpha\right) + \left(i + \alpha\right) \cdot \left(i + \alpha\right)\right)}{t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{{\left(\beta + i\right)}^{2}}}}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 6.1 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))) (t_1 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 4.2e+58)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (*
        (/ i (fma t_0 t_0 -1.0))
        (/
         (+
          (+ (* i (pow (+ i alpha) 2.0)) (* (* beta beta) (+ i alpha)))
          (* beta (+ (* i (+ i alpha)) (* (+ i alpha) (+ i alpha)))))
         (* t_0 t_0)))
       (if (<= beta 1.7e+97)
         0.0625
         (if (<= beta 1.6e+141)
           (/
            (/
             (* i i)
             (/
              (fma 4.0 (* beta i) (fma 4.0 (* i i) (* beta beta)))
              (pow (+ beta i) 2.0)))
            (+ -1.0 (* t_1 t_1)))
           (if (<= beta 6.1e+170)
             (-
              (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
              (* 0.125 (/ (+ beta alpha) i)))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4.2e+58) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = (i / fma(t_0, t_0, -1.0)) * ((((i * pow((i + alpha), 2.0)) + ((beta * beta) * (i + alpha))) + (beta * ((i * (i + alpha)) + ((i + alpha) * (i + alpha))))) / (t_0 * t_0));
	} else if (beta <= 1.7e+97) {
		tmp = 0.0625;
	} else if (beta <= 1.6e+141) {
		tmp = ((i * i) / (fma(4.0, (beta * i), fma(4.0, (i * i), (beta * beta))) / pow((beta + i), 2.0))) / (-1.0 + (t_1 * t_1));
	} else if (beta <= 6.1e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4.2e+58)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i / fma(t_0, t_0, -1.0)) * Float64(Float64(Float64(Float64(i * (Float64(i + alpha) ^ 2.0)) + Float64(Float64(beta * beta) * Float64(i + alpha))) + Float64(beta * Float64(Float64(i * Float64(i + alpha)) + Float64(Float64(i + alpha) * Float64(i + alpha))))) / Float64(t_0 * t_0)));
	elseif (beta <= 1.7e+97)
		tmp = 0.0625;
	elseif (beta <= 1.6e+141)
		tmp = Float64(Float64(Float64(i * i) / Float64(fma(4.0, Float64(beta * i), fma(4.0, Float64(i * i), Float64(beta * beta))) / (Float64(beta + i) ^ 2.0))) / Float64(-1.0 + Float64(t_1 * t_1)));
	elseif (beta <= 6.1e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.2e+58], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(i * N[Power[N[(i + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(i + alpha), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.7e+97], 0.0625, If[LessEqual[beta, 1.6e+141], N[(N[(N[(i * i), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.1e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if beta < 4.20000000000000024e58

   1. Initial program 21.8%

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

      1. associate-/l/ 20.5%

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

      2. associate-*l* 20.4%

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

      3. times-frac 26.6%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in alpha around 0 18.5%

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

      1. times-frac 41.1%

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

      2. unpow2 41.1%

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

      3. +-commutative 41.1%

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

      4. *-commutative 41.1%

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

      5. fma-udef 41.1%

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

      6. sub-neg 41.1%

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

      7. +-commutative 41.1%

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

      8. *-commutative 41.1%

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

      9. fma-udef 41.1%

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

      10. metadata-eval 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in i around inf 77.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 77.4%

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

      2. associate-*r/ 77.4%

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

      3. unpow2 77.4%

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

      4. unpow2 77.4%

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

      5. associate-*r/ 77.4%

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

   9. Simplified 77.4%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.20000000000000024e58 < beta < 3.2999999999999999e69

   1. Initial program 59.3%

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

      1. associate-/l/ 59.3%

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

      2. associate-*l* 59.3%

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

      3. times-frac 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in beta around -inf 59.4%

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

      1. +-commutative 59.4%

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

      2. mul-1-neg 59.4%

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

      3. unsub-neg 59.4%

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

   6. Simplified 59.4%

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

   if 3.2999999999999999e69 < beta < 1.70000000000000005e97

   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. times-frac 0.8%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in i around inf 74.4%

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

   if 1.70000000000000005e97 < beta < 1.60000000000000009e141

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around 0 15.9%

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

      1. associate-/l* 84.9%

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

      2. unpow2 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. fma-udef 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in i around 0 85.3%

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

      1. fma-def 85.3%

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

      2. fma-def 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

   7. Simplified 85.3%

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

   if 1.60000000000000009e141 < beta < 6.1000000000000004e170

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in i around inf 84.4%

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

   5. Taylor expanded in alpha around 0 76.2%

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

   if 6.1000000000000004e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+58}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)} \cdot \frac{\left(i \cdot {\left(i + \alpha\right)}^{2} + \left(\beta \cdot \beta\right) \cdot \left(i + \alpha\right)\right) + \beta \cdot \left(i \cdot \left(i + \alpha\right) + \left(i + \alpha\right) \cdot \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 6.1 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := -1 + t_0 \cdot t_0\\ t_2 := {\left(\beta + i\right)}^{2}\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{t_2 \cdot {i}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{t_1}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{t_2}}}{t_1}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (+ -1.0 (* t_0 t_0)))
        (t_2 (pow (+ beta i) 2.0)))
   (if (<= beta 4.1e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (/ (* t_2 (pow i 2.0)) (pow (+ beta (* i 2.0)) 2.0)) t_1)
       (if (<= beta 4e+99)
         0.0625
         (if (<= beta 5.3e+141)
           (/
            (/
             (* i i)
             (/ (fma 4.0 (* beta i) (fma 4.0 (* i i) (* beta beta))) t_2))
            t_1)
           (if (<= beta 4.3e+170)
             (-
              (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
              (* 0.125 (/ (+ beta alpha) i)))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = -1.0 + (t_0 * t_0);
	double t_2 = pow((beta + i), 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((t_2 * pow(i, 2.0)) / pow((beta + (i * 2.0)), 2.0)) / t_1;
	} else if (beta <= 4e+99) {
		tmp = 0.0625;
	} else if (beta <= 5.3e+141) {
		tmp = ((i * i) / (fma(4.0, (beta * i), fma(4.0, (i * i), (beta * beta))) / t_2)) / t_1;
	} else if (beta <= 4.3e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(-1.0 + Float64(t_0 * t_0))
	t_2 = Float64(beta + i) ^ 2.0
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(t_2 * (i ^ 2.0)) / (Float64(beta + Float64(i * 2.0)) ^ 2.0)) / t_1);
	elseif (beta <= 4e+99)
		tmp = 0.0625;
	elseif (beta <= 5.3e+141)
		tmp = Float64(Float64(Float64(i * i) / Float64(fma(4.0, Float64(beta * i), fma(4.0, Float64(i * i), Float64(beta * beta))) / t_2)) / t_1);
	elseif (beta <= 4.3e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(t$95$2 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[beta, 4e+99], 0.0625, If[LessEqual[beta, 5.3e+141], N[(N[(N[(i * i), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[beta, 4.3e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   2. Taylor expanded in alpha around 0 99.4%

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

   if 3.2999999999999999e69 < beta < 3.9999999999999999e99

   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. times-frac 0.8%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in i around inf 74.4%

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

   if 3.9999999999999999e99 < beta < 5.3e141

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around 0 15.9%

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

      1. associate-/l* 84.9%

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

      2. unpow2 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. fma-udef 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in i around 0 85.3%

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

      1. fma-def 85.3%

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

      2. fma-def 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

   7. Simplified 85.3%

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

   if 5.3e141 < beta < 4.2999999999999999e170

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in i around inf 84.4%

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

   5. Taylor expanded in alpha around 0 76.2%

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

   if 4.2999999999999999e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{{\left(\beta + i\right)}^{2} \cdot {i}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \mathsf{fma}\left(4, i \cdot i, \beta \cdot \beta\right)\right)}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + i \cdot 2\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := {\left(\beta + i\right)}^{2}\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot t_2}{\mathsf{fma}\left(t_0, t_0, -1\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;\beta \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{t_2}}}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 4.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta (* i 2.0))))
        (t_1 (+ (+ beta alpha) (* i 2.0)))
        (t_2 (pow (+ beta i) 2.0)))
   (if (<= beta 4.1e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (* (* i i) t_2) (* (fma t_0 t_0 -1.0) (* t_0 t_0)))
       (if (<= beta 1.25e+99)
         0.0625
         (if (<= beta 1.4e+141)
           (/
            (/ (* i i) (/ (pow (fma i 2.0 beta) 2.0) t_2))
            (+ -1.0 (* t_1 t_1)))
           (if (<= beta 4.9e+170)
             (-
              (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
              (* 0.125 (/ (+ beta alpha) i)))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + (beta + (i * 2.0));
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = pow((beta + i), 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((i * i) * t_2) / (fma(t_0, t_0, -1.0) * (t_0 * t_0));
	} else if (beta <= 1.25e+99) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+141) {
		tmp = ((i * i) / (pow(fma(i, 2.0, beta), 2.0) / t_2)) / (-1.0 + (t_1 * t_1));
	} else if (beta <= 4.9e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(beta + Float64(i * 2.0)))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_2 = Float64(beta + i) ^ 2.0
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(i * i) * t_2) / Float64(fma(t_0, t_0, -1.0) * Float64(t_0 * t_0)));
	elseif (beta <= 1.25e+99)
		tmp = 0.0625;
	elseif (beta <= 1.4e+141)
		tmp = Float64(Float64(Float64(i * i) / Float64((fma(i, 2.0, beta) ^ 2.0) / t_2)) / Float64(-1.0 + Float64(t_1 * t_1)));
	elseif (beta <= 4.9e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(i * i), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.25e+99], 0.0625, If[LessEqual[beta, 1.4e+141], N[(N[(N[(i * i), $MachinePrecision] / N[(N[Power[N[(i * 2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.9e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in alpha around 0 99.2%

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

      1. unpow2 99.2%

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

   6. Simplified 99.2%

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

   if 3.2999999999999999e69 < beta < 1.25000000000000002e99

   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. times-frac 0.8%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in i around inf 74.4%

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

   if 1.25000000000000002e99 < beta < 1.39999999999999996e141

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around 0 15.9%

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

      1. associate-/l* 84.9%

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

      2. unpow2 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. fma-udef 84.9%

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

   4. Simplified 84.9%

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

   if 1.39999999999999996e141 < beta < 4.9000000000000004e170

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in i around inf 84.4%

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

   5. Taylor expanded in alpha around 0 76.2%

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

   if 4.9000000000000004e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := -1 + t_0 \cdot t_0\\ t_2 := {\left(\beta + i\right)}^{2}\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{t_2 \cdot {i}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{t_1}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{t_2}}}{t_1}\\ \mathbf{elif}\;\beta \leq 4.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (+ -1.0 (* t_0 t_0)))
        (t_2 (pow (+ beta i) 2.0)))
   (if (<= beta 4.1e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (/ (* t_2 (pow i 2.0)) (pow (+ beta (* i 2.0)) 2.0)) t_1)
       (if (<= beta 4.3e+99)
         0.0625
         (if (<= beta 1.55e+142)
           (/ (/ (* i i) (/ (pow (fma i 2.0 beta) 2.0) t_2)) t_1)
           (if (<= beta 4.9e+170)
             (-
              (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
              (* 0.125 (/ (+ beta alpha) i)))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = -1.0 + (t_0 * t_0);
	double t_2 = pow((beta + i), 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((t_2 * pow(i, 2.0)) / pow((beta + (i * 2.0)), 2.0)) / t_1;
	} else if (beta <= 4.3e+99) {
		tmp = 0.0625;
	} else if (beta <= 1.55e+142) {
		tmp = ((i * i) / (pow(fma(i, 2.0, beta), 2.0) / t_2)) / t_1;
	} else if (beta <= 4.9e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(-1.0 + Float64(t_0 * t_0))
	t_2 = Float64(beta + i) ^ 2.0
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(t_2 * (i ^ 2.0)) / (Float64(beta + Float64(i * 2.0)) ^ 2.0)) / t_1);
	elseif (beta <= 4.3e+99)
		tmp = 0.0625;
	elseif (beta <= 1.55e+142)
		tmp = Float64(Float64(Float64(i * i) / Float64((fma(i, 2.0, beta) ^ 2.0) / t_2)) / t_1);
	elseif (beta <= 4.9e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(t$95$2 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[beta, 4.3e+99], 0.0625, If[LessEqual[beta, 1.55e+142], N[(N[(N[(i * i), $MachinePrecision] / N[(N[Power[N[(i * 2.0 + beta), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[beta, 4.9e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   2. Taylor expanded in alpha around 0 99.4%

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

   if 3.2999999999999999e69 < beta < 4.3000000000000001e99

   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. times-frac 0.8%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in i around inf 74.4%

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

   if 4.3000000000000001e99 < beta < 1.55e142

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around 0 15.9%

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

      1. associate-/l* 84.9%

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

      2. unpow2 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. fma-udef 84.9%

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

   4. Simplified 84.9%

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

   if 1.55e142 < beta < 4.9000000000000004e170

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in i around inf 84.4%

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

   5. Taylor expanded in alpha around 0 76.2%

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

   if 4.9000000000000004e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{{\left(\beta + i\right)}^{2} \cdot {i}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + i \cdot 2\right)\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\mathsf{fma}\left(t_0, t_0, -1\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta (* i 2.0)))))
   (if (<= beta 4.1e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (* (* i i) (pow (+ beta i) 2.0)) (* (fma t_0 t_0 -1.0) (* t_0 t_0)))
       (if (<= beta 3.5e+171)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + (beta + (i * 2.0));
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((i * i) * pow((beta + i), 2.0)) / (fma(t_0, t_0, -1.0) * (t_0 * t_0));
	} else if (beta <= 3.5e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(beta + Float64(i * 2.0)))
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(i * i) * (Float64(beta + i) ^ 2.0)) / Float64(fma(t_0, t_0, -1.0) * Float64(t_0 * t_0)));
	elseif (beta <= 3.5e+171)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(i * i), $MachinePrecision] * N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.5e+171], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in alpha around 0 99.2%

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

      1. unpow2 99.2%

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

   6. Simplified 99.2%

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

   if 3.2999999999999999e69 < beta < 3.4999999999999999e171

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 3.4999999999999999e171 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 7: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + i \cdot 2\right)\\ t_1 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_1 \cdot \left(t_1 + \beta \cdot \alpha\right)}{\mathsf{fma}\left(t_0, t_0, -1\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;\beta \leq 8.4 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta (* i 2.0)))) (t_1 (* i (+ i (+ beta alpha)))))
   (if (<= beta 2.85e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (* t_1 (+ t_1 (* beta alpha))) (* (fma t_0 t_0 -1.0) (* t_0 t_0)))
       (if (<= beta 8.4e+170)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + (beta + (i * 2.0));
	double t_1 = i * (i + (beta + alpha));
	double tmp;
	if (beta <= 2.85e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = (t_1 * (t_1 + (beta * alpha))) / (fma(t_0, t_0, -1.0) * (t_0 * t_0));
	} else if (beta <= 8.4e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(beta + Float64(i * 2.0)))
	t_1 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 2.85e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(t_1 * Float64(t_1 + Float64(beta * alpha))) / Float64(fma(t_0, t_0, -1.0) * Float64(t_0 * t_0)));
	elseif (beta <= 8.4e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.85e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(t$95$1 * N[(t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8.4e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.8500000000000001e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 2.8500000000000001e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*l* 74.1%

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

      2. +-commutative 74.1%

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

      3. fma-udef 74.1%

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

      4. *-commutative 74.1%

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

      5. +-commutative 74.1%

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

      6. associate-*l* 74.1%

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

      7. distribute-rgt-in 74.1%

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

      8. *-commutative 74.1%

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

      9. +-commutative 74.1%

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

      10. +-commutative 74.1%

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

      11. +-commutative 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. distribute-rgt-out 74.1%

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

      2. +-commutative 74.1%

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

      3. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   if 3.2999999999999999e69 < beta < 8.39999999999999991e170

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 8.39999999999999991e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\beta \leq 8.4 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{t_2 \cdot \left(t_2 + \beta \cdot \alpha\right)}{t_1}}{-1 + t_1}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha)))))
   (if (<= beta 4e+63)
     (+
      0.0625
      (-
       (/ (* 0.0625 (* beta beta)) (* i i))
       (/
        (* 0.00390625 (fma 4.0 (+ (* beta beta) -1.0) (* (* beta beta) 20.0)))
        (* i i))))
     (if (<= beta 3.3e+69)
       (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1))
       (if (<= beta 7.4e+171)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double tmp;
	if (beta <= 4e+63) {
		tmp = 0.0625 + (((0.0625 * (beta * beta)) / (i * i)) - ((0.00390625 * fma(4.0, ((beta * beta) + -1.0), ((beta * beta) * 20.0))) / (i * i)));
	} else if (beta <= 3.3e+69) {
		tmp = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1);
	} else if (beta <= 7.4e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 4e+63)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(beta * beta)) / Float64(i * i)) - Float64(Float64(0.00390625 * fma(4.0, Float64(Float64(beta * beta) + -1.0), Float64(Float64(beta * beta) * 20.0))) / Float64(i * i))));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1));
	elseif (beta <= 7.4e+171)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+63], N[(0.0625 + N[(N[(N[(0.0625 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(0.00390625 * N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 7.4e+171], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.00000000000000023e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   if 4.00000000000000023e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   if 3.2999999999999999e69 < beta < 7.39999999999999996e171

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 7.39999999999999996e171 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := t_1 \cdot t_1\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{t_0 \cdot \left(t_0 + \beta \cdot \alpha\right)}{t_2}}{-1 + t_2}\\ \mathbf{elif}\;\beta \leq 4.7 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ i (+ beta alpha))))
        (t_1 (+ (+ beta alpha) (* i 2.0)))
        (t_2 (* t_1 t_1)))
   (if (<= beta 4.1e+63)
     (+ 0.0625 (/ 0.015625 (* i i)))
     (if (<= beta 3.3e+69)
       (/ (/ (* t_0 (+ t_0 (* beta alpha))) t_2) (+ -1.0 t_2))
       (if (<= beta 4.7e+170)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = i * (i + (beta + alpha));
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = ((t_0 * (t_0 + (beta * alpha))) / t_2) / (-1.0 + t_2);
	} else if (beta <= 4.7e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = i * (i + (beta + alpha))
    t_1 = (beta + alpha) + (i * 2.0d0)
    t_2 = t_1 * t_1
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = ((t_0 * (t_0 + (beta * alpha))) / t_2) / ((-1.0d0) + t_2)
    else if (beta <= 4.7d+170) then
        tmp = (0.0625d0 + (0.0625d0 * (2.0d0 * (beta / i)))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + (beta + alpha));
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = ((t_0 * (t_0 + (beta * alpha))) / t_2) / (-1.0 + t_2);
	} else if (beta <= 4.7e+170) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = i * (i + (beta + alpha))
	t_1 = (beta + alpha) + (i * 2.0)
	t_2 = t_1 * t_1
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = ((t_0 * (t_0 + (beta * alpha))) / t_2) / (-1.0 + t_2)
	elif beta <= 4.7e+170:
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(i * Float64(i + Float64(beta + alpha)))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(t_0 * Float64(t_0 + Float64(beta * alpha))) / t_2) / Float64(-1.0 + t_2));
	elseif (beta <= 4.7e+170)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + (beta + alpha));
	t_1 = (beta + alpha) + (i * 2.0);
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = ((t_0 * (t_0 + (beta * alpha))) / t_2) / (-1.0 + t_2);
	elseif (beta <= 4.7e+170)
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(t$95$0 * N[(t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.7e+170], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   if 3.2999999999999999e69 < beta < 4.70000000000000004e170

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 4.70000000000000004e170 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 4.7 \cdot 10^{+170}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 10: 84.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{i \cdot i}{1 - \frac{i \cdot -2}{\beta}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 4.1e+63)
     (+ 0.0625 (/ 0.015625 (* i i)))
     (if (<= beta 3.3e+69)
       (/ (/ (* i i) (- 1.0 (/ (* i -2.0) beta))) (+ -1.0 (* t_0 t_0)))
       (if (<= beta 3.6e+171)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = ((i * i) / (1.0 - ((i * -2.0) / beta))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + (i * 2.0d0)
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = ((i * i) / (1.0d0 - ((i * (-2.0d0)) / beta))) / ((-1.0d0) + (t_0 * t_0))
    else if (beta <= 3.6d+171) then
        tmp = (0.0625d0 + (0.0625d0 * (2.0d0 * (beta / i)))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = ((i * i) / (1.0 - ((i * -2.0) / beta))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.6e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = ((i * i) / (1.0 - ((i * -2.0) / beta))) / (-1.0 + (t_0 * t_0))
	elif beta <= 3.6e+171:
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(Float64(i * i) / Float64(1.0 - Float64(Float64(i * -2.0) / beta))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 3.6e+171)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = ((i * i) / (1.0 - ((i * -2.0) / beta))) / (-1.0 + (t_0 * t_0));
	elseif (beta <= 3.6e+171)
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(N[(i * i), $MachinePrecision] / N[(1.0 - N[(N[(i * -2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.6e+171], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   2. Taylor expanded in alpha around 0 99.4%

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

      1. associate-/l* 99.1%

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

      2. unpow2 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in beta around -inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. unsub-neg 78.4%

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

      3. distribute-rgt-out-- 78.4%

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

      4. metadata-eval 78.4%

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

   7. Simplified 78.4%

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

   if 3.2999999999999999e69 < beta < 3.60000000000000018e171

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 3.60000000000000018e171 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{i \cdot i}{1 - \frac{i \cdot -2}{\beta}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 11: 84.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 4.1e+63)
     (+ 0.0625 (/ 0.015625 (* i i)))
     (if (<= beta 3.3e+69)
       (/ (* i i) (+ -1.0 (* t_0 t_0)))
       (if (<= beta 6.4e+171)
         (-
          (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
          (* 0.125 (/ (+ beta alpha) i)))
         (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 6.4e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + (i * 2.0d0)
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = (i * i) / ((-1.0d0) + (t_0 * t_0))
    else if (beta <= 6.4d+171) then
        tmp = (0.0625d0 + (0.0625d0 * (2.0d0 * (beta / i)))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 6.4e+171) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = (i * i) / (-1.0 + (t_0 * t_0))
	elif beta <= 6.4e+171:
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i * i) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 6.4e+171)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = (i * i) / (-1.0 + (t_0 * t_0));
	elseif (beta <= 6.4e+171)
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i * i), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.4e+171], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

   2. Taylor expanded in alpha around 0 99.4%

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

      1. associate-/l* 99.1%

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

      2. unpow2 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in beta around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. distribute-lft-out 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. fma-def 75.4%

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

      6. +-commutative 75.4%

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

      7. unpow2 75.4%

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

      8. unpow2 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in i around 0 79.3%

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

      1. unpow2 79.3%

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

   10. Simplified 79.3%

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

   if 3.2999999999999999e69 < beta < 6.40000000000000022e171

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 6.40000000000000022e171 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+171}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 12: 84.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.7e+63)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 3.3e+69)
     (/ (* i i) (* beta beta))
     (if (<= beta 2e+172)
       (-
        (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
        (* 0.125 (/ (+ beta alpha) i)))
       (* (/ i beta) (/ (+ i alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.7e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2e+172) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.7d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = (i * i) / (beta * beta)
    else if (beta <= 2d+172) then
        tmp = (0.0625d0 + (0.0625d0 * (2.0d0 * (beta / i)))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.7e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2e+172) {
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.7e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = (i * i) / (beta * beta)
	elif beta <= 2e+172:
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.7e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	elseif (beta <= 2e+172)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.7e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = (i * i) / (beta * beta);
	elseif (beta <= 2e+172)
		tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.7e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2e+172], N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 3.69999999999999968e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 3.69999999999999968e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

      1. associate-/l/ 74.1%

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

      2. associate-*l* 74.1%

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

      3. times-frac 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in beta around inf 53.2%

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

      1. associate-/l* 52.9%

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

      2. unpow2 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in i around inf 53.7%

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

      1. unpow2 53.7%

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

      2. unpow2 53.7%

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

   9. Simplified 53.7%

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

   if 3.2999999999999999e69 < beta < 2.0000000000000002e172

   1. Initial program 4.1%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.8%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in i around inf 71.0%

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

   5. Taylor expanded in alpha around 0 64.1%

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

   if 2.0000000000000002e172 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. associate-/l* 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

      1. div-inv 16.3%

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

   8. Applied egg-rr 16.3%

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

   9. Taylor expanded in beta around 0 13.6%

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

       1. +-commutative 13.6%

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

       2. unpow2 13.6%

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

       3. times-frac 74.6%

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

       4. +-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 13: 84.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+63)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 3.3e+69)
     (/ (* i i) (* beta beta))
     (if (<= beta 2.3e+173) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.3e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = (i * i) / (beta * beta)
    else if (beta <= 2.3d+173) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.3e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = (i * i) / (beta * beta)
	elif beta <= 2.3e+173:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	elseif (beta <= 2.3e+173)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = (i * i) / (beta * beta);
	elseif (beta <= 2.3e+173)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.3e+173], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

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

      2. associate-*l* 20.3%

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

      3. times-frac 26.5%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in alpha around 0 18.4%

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

      1. times-frac 40.9%

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

      2. unpow2 40.9%

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

      3. +-commutative 40.9%

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

      4. *-commutative 40.9%

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

      5. fma-udef 40.9%

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

      6. sub-neg 40.9%

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

      7. +-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. fma-udef 40.9%

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

      10. metadata-eval 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r/ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

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

      1. associate-/l/ 74.1%

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

      2. associate-*l* 74.1%

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

      3. times-frac 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in beta around inf 53.2%

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

      1. associate-/l* 52.9%

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

      2. unpow2 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in i around inf 53.7%

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

      1. unpow2 53.7%

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

      2. unpow2 53.7%

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

   9. Simplified 53.7%

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

   if 3.2999999999999999e69 < beta < 2.29999999999999995e173

   1. Initial program 4.0%

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

      1. associate-/l/ 0.1%

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

      2. associate-*l* 0.1%

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

      3. times-frac 13.4%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in i around inf 65.6%

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

   if 2.29999999999999995e173 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 8.1%

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

   4. Taylor expanded in beta around inf 14.0%

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

      1. associate-/l* 16.7%

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

      2. unpow2 16.7%

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

   6. Simplified 16.7%

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

      1. div-inv 16.7%

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

   8. Applied egg-rr 16.7%

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

   9. Taylor expanded in beta around 0 14.0%

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

       1. +-commutative 14.0%

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

       2. unpow2 14.0%

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

       3. times-frac 76.4%

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

       4. +-commutative 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 14: 84.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.3e+61)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 3.3e+69)
     (/ (* i i) (* beta beta))
     (if (<= beta 2.4e+173) 0.0625 (/ (* (/ i beta) (+ i alpha)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.3e+61) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.4e+173) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.3d+61) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = (i * i) / (beta * beta)
    else if (beta <= 2.4d+173) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * (i + alpha)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.3e+61) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.4e+173) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.3e+61:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = (i * i) / (beta * beta)
	elif beta <= 2.4e+173:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * (i + alpha)) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.3e+61)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	elseif (beta <= 2.4e+173)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(i + alpha)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.3e+61)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = (i * i) / (beta * beta);
	elseif (beta <= 2.4e+173)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * (i + alpha)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.3e+61], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.4e+173], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.29999999999999986e61

   1. Initial program 21.7%

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

      1. associate-/l/ 20.4%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 20.3%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 26.5%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in alpha around 0 18.4%

      \[\leadsto \color{blue}{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} \cdot \left({\left(\beta + 2 \cdot i\right)}^{2} - 1\right)}} \]
   5. Step-by-step derivation

      1. times-frac 40.9%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1}} \]

      2. unpow2 40.9%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      3. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      4. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      5. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      6. sub-neg 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{2} + \left(-1\right)}} \]

      7. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2} + \left(-1\right)} \]

      8. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2} + \left(-1\right)} \]

      9. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2} + \left(-1\right)} \]

      10. metadata-eval 40.9%

          \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + \color{blue}{-1}} \]

   6. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}} \]

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

         \[\leadsto \color{blue}{0.0625 + \left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

      2. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\color{blue}{\frac{0.0625 \cdot {\beta}^{2}}{{i}^{2}}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      3. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      4. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      5. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \color{blue}{\frac{0.00390625 \cdot \left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right)}{{i}^{2}}}\right) \]

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 1.29999999999999986e61 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 74.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 74.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 73.7%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 53.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 52.9%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 52.9%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

   7. Taylor expanded in i around inf 53.7%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 53.7%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 53.7%

         \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{i \cdot i}{\beta \cdot \beta}} \]

   if 3.2999999999999999e69 < beta < 2.3999999999999999e173

   1. Initial program 4.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 13.4%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 65.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.3999999999999999e173 < beta

   1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.0%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 0.0%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 14.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 16.7%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 16.7%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 16.7%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 16.7%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 16.7%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in beta around 0 14.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative 14.0%

          \[\leadsto \frac{i \cdot \color{blue}{\left(i + \alpha\right)}}{{\beta}^{2}} \]

       2. unpow2 14.0%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 76.4%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

       4. +-commutative 76.4%

          \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{\alpha + i}}{\beta} \]

   11. Simplified 76.4%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]
   12. Step-by-step derivation

       1. associate-*r/ 76.3%

          \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\beta}} \]

       2. +-commutative 76.3%

          \[\leadsto \frac{\frac{i}{\beta} \cdot \color{blue}{\left(i + \alpha\right)}}{\beta} \]

   13. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \]

Alternative 15: 82.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69} \lor \neg \left(\beta \leq 2 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+63)
   0.0625
   (if (or (<= beta 3.3e+69) (not (<= beta 2e+173)))
     (* (/ i beta) (/ i beta))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625;
	} else if ((beta <= 3.3e+69) || !(beta <= 2e+173)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0
    else if ((beta <= 3.3d+69) .or. (.not. (beta <= 2d+173))) then
        tmp = (i / beta) * (i / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625;
	} else if ((beta <= 3.3e+69) || !(beta <= 2e+173)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625
	elif (beta <= 3.3e+69) or not (beta <= 2e+173):
		tmp = (i / beta) * (i / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = 0.0625;
	elseif ((beta <= 3.3e+69) || !(beta <= 2e+173))
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625;
	elseif ((beta <= 3.3e+69) || ~((beta <= 2e+173)))
		tmp = (i / beta) * (i / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.1e+63], 0.0625, If[Or[LessEqual[beta, 3.3e+69], N[Not[LessEqual[beta, 2e+173]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.09999999999999993e63 or 3.2999999999999999e69 < beta < 2e173

   1. Initial program 19.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 17.5%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 17.3%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 24.6%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 75.0%

      \[\leadsto \color{blue}{0.0625} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69 or 2e173 < beta

   1. Initial program 6.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 6.7%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 6.7%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 6.7%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 17.5%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 20.0%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 20.0%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 20.0%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 20.0%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 20.0%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in i around inf 17.7%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 17.7%

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

       2. unpow2 17.7%

          \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 68.0%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69} \lor \neg \left(\beta \leq 2 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 16: 83.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69} \lor \neg \left(\beta \leq 3.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+63)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (or (<= beta 3.3e+69) (not (<= beta 3.5e+173)))
     (* (/ i beta) (/ i beta))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if ((beta <= 3.3e+69) || !(beta <= 3.5e+173)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if ((beta <= 3.3d+69) .or. (.not. (beta <= 3.5d+173))) then
        tmp = (i / beta) * (i / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if ((beta <= 3.3e+69) || !(beta <= 3.5e+173)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif (beta <= 3.3e+69) or not (beta <= 3.5e+173):
		tmp = (i / beta) * (i / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif ((beta <= 3.3e+69) || !(beta <= 3.5e+173))
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif ((beta <= 3.3e+69) || ~((beta <= 3.5e+173)))
		tmp = (i / beta) * (i / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[beta, 3.3e+69], N[Not[LessEqual[beta, 3.5e+173]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 20.4%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 20.3%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 26.5%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in alpha around 0 18.4%

      \[\leadsto \color{blue}{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} \cdot \left({\left(\beta + 2 \cdot i\right)}^{2} - 1\right)}} \]
   5. Step-by-step derivation

      1. times-frac 40.9%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1}} \]

      2. unpow2 40.9%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      3. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      4. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      5. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      6. sub-neg 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{2} + \left(-1\right)}} \]

      7. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2} + \left(-1\right)} \]

      8. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2} + \left(-1\right)} \]

      9. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2} + \left(-1\right)} \]

      10. metadata-eval 40.9%

          \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + \color{blue}{-1}} \]

   6. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}} \]

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

         \[\leadsto \color{blue}{0.0625 + \left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

      2. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\color{blue}{\frac{0.0625 \cdot {\beta}^{2}}{{i}^{2}}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      3. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      4. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      5. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \color{blue}{\frac{0.00390625 \cdot \left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right)}{{i}^{2}}}\right) \]

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69 or 3.4999999999999999e173 < beta

   1. Initial program 6.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 6.7%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 6.7%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 6.7%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 17.5%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 20.0%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 20.0%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 20.0%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 20.0%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 20.0%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in i around inf 17.7%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 17.7%

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

       2. unpow2 17.7%

          \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 68.0%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

   11. Simplified 68.0%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

   if 3.2999999999999999e69 < beta < 3.4999999999999999e173

   1. Initial program 4.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 13.4%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 65.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69} \lor \neg \left(\beta \leq 3.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 17: 83.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.36 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+63)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 3.3e+69)
     (/ i (* beta (/ beta i)))
     (if (<= beta 2.36e+173) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = i / (beta * (beta / i));
	} else if (beta <= 2.36e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = i / (beta * (beta / i))
    else if (beta <= 2.36d+173) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = i / (beta * (beta / i));
	} else if (beta <= 2.36e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = i / (beta * (beta / i))
	elif beta <= 2.36e+173:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(i / Float64(beta * Float64(beta / i)));
	elseif (beta <= 2.36e+173)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = i / (beta * (beta / i));
	elseif (beta <= 2.36e+173)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(i / N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.36e+173], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 20.4%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 20.3%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 26.5%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in alpha around 0 18.4%

      \[\leadsto \color{blue}{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} \cdot \left({\left(\beta + 2 \cdot i\right)}^{2} - 1\right)}} \]
   5. Step-by-step derivation

      1. times-frac 40.9%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1}} \]

      2. unpow2 40.9%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      3. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      4. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      5. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      6. sub-neg 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{2} + \left(-1\right)}} \]

      7. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2} + \left(-1\right)} \]

      8. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2} + \left(-1\right)} \]

      9. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2} + \left(-1\right)} \]

      10. metadata-eval 40.9%

          \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + \color{blue}{-1}} \]

   6. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}} \]

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

         \[\leadsto \color{blue}{0.0625 + \left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

      2. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\color{blue}{\frac{0.0625 \cdot {\beta}^{2}}{{i}^{2}}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      3. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      4. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      5. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \color{blue}{\frac{0.00390625 \cdot \left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right)}{{i}^{2}}}\right) \]

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 74.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 74.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 73.7%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 53.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 52.9%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 52.9%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 52.9%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 52.9%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in alpha around 0 53.3%

      \[\leadsto \frac{i}{\color{blue}{\frac{{\beta}^{2}}{i}}} \]
   10. Step-by-step derivation

       1. unpow2 53.3%

          \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

       2. associate-/l* 53.3%

          \[\leadsto \frac{i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

       3. associate-/r/ 53.3%

          \[\leadsto \frac{i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]

   11. Simplified 53.3%

       \[\leadsto \frac{i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]

   if 3.2999999999999999e69 < beta < 2.3599999999999999e173

   1. Initial program 4.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 13.4%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 65.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.3599999999999999e173 < beta

   1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.0%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 0.0%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 14.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 16.7%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 16.7%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 16.7%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 16.7%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 16.7%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in i around inf 14.2%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 14.2%

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

       2. unpow2 14.2%

          \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 69.5%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

   11. Simplified 69.5%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.36 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 18: 83.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+63)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 3.3e+69)
     (/ (* i i) (* beta beta))
     (if (<= beta 2.45e+173) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.45e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.1d+63) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 3.3d+69) then
        tmp = (i * i) / (beta * beta)
    else if (beta <= 2.45d+173) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+63) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 3.3e+69) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 2.45e+173) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+63:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 3.3e+69:
		tmp = (i * i) / (beta * beta)
	elif beta <= 2.45e+173:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+63)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 3.3e+69)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	elseif (beta <= 2.45e+173)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+63)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 3.3e+69)
		tmp = (i * i) / (beta * beta);
	elseif (beta <= 2.45e+173)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.1e+63], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.3e+69], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.45e+173], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.09999999999999993e63

   1. Initial program 21.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 20.4%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 20.3%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 26.5%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in alpha around 0 18.4%

      \[\leadsto \color{blue}{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} \cdot \left({\left(\beta + 2 \cdot i\right)}^{2} - 1\right)}} \]
   5. Step-by-step derivation

      1. times-frac 40.9%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1}} \]

      2. unpow2 40.9%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      3. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      4. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      5. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]

      6. sub-neg 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{2} + \left(-1\right)}} \]

      7. +-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2} + \left(-1\right)} \]

      8. *-commutative 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2} + \left(-1\right)} \]

      9. fma-udef 40.9%

         \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2} + \left(-1\right)} \]

      10. metadata-eval 40.9%

          \[\leadsto \frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + \color{blue}{-1}} \]

   6. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}} \]

   7. Taylor expanded in i around inf 76.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. associate--l+ 76.9%

         \[\leadsto \color{blue}{0.0625 + \left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

      2. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\color{blue}{\frac{0.0625 \cdot {\beta}^{2}}{{i}^{2}}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      3. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      4. unpow2 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}} - 0.00390625 \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right) \]

      5. associate-*r/ 76.9%

         \[\leadsto 0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \color{blue}{\frac{0.00390625 \cdot \left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right)}{{i}^{2}}}\right) \]

   9. Simplified 76.9%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i} - \frac{0.00390625 \cdot \mathsf{fma}\left(4, \beta \cdot \beta + -1, \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}\right)} \]

   10. Taylor expanded in beta around 0 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{{i}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 77.0%

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   12. Simplified 77.0%

       \[\leadsto 0.0625 + \color{blue}{\frac{0.015625}{i \cdot i}} \]

   if 4.09999999999999993e63 < beta < 3.2999999999999999e69

   1. Initial program 74.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 74.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 74.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 73.7%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 53.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 52.9%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 52.9%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]

   7. Taylor expanded in i around inf 53.7%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 53.7%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 53.7%

         \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{i \cdot i}{\beta \cdot \beta}} \]

   if 3.2999999999999999e69 < beta < 2.45e173

   1. Initial program 4.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.1%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 13.4%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 65.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.45e173 < beta

   1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.0%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 0.0%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 14.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 16.7%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 16.7%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 16.7%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 16.7%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 16.7%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in i around inf 14.2%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 14.2%

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

       2. unpow2 14.2%

          \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 69.5%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

   11. Simplified 69.5%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+63}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 19: 75.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.25e+198) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+198) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.25d+198) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+198) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.25e+198:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.25e+198)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.25e+198)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.25e+198], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25000000000000001e198

   1. Initial program 19.7%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 18.2%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 18.1%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 25.0%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in i around inf 72.7%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.25000000000000001e198 < beta

   1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l* 0.0%

         \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac 0.0%

         \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

   4. Taylor expanded in beta around inf 15.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/l* 18.0%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha + i}}} \]

      2. unpow2 18.0%

         \[\leadsto \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{\alpha + i}} \]

   6. Simplified 18.0%

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}}} \]
   7. Step-by-step derivation

      1. div-inv 18.0%

         \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   8. Applied egg-rr 18.0%

      \[\leadsto \frac{i}{\color{blue}{\left(\beta \cdot \beta\right) \cdot \frac{1}{\alpha + i}}} \]

   9. Taylor expanded in beta around 0 15.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative 15.0%

          \[\leadsto \frac{i \cdot \color{blue}{\left(i + \alpha\right)}}{{\beta}^{2}} \]

       2. unpow2 15.0%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 73.8%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

       4. +-commutative 73.8%

          \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{\alpha + i}}{\beta} \]

   11. Simplified 73.8%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]

   12. Taylor expanded in i around 0 17.2%

       \[\leadsto \color{blue}{\frac{\alpha \cdot i}{{\beta}^{2}}} \]
   13. Step-by-step derivation

       1. *-commutative 17.2%

          \[\leadsto \frac{\color{blue}{i \cdot \alpha}}{{\beta}^{2}} \]

       2. unpow2 17.2%

          \[\leadsto \frac{i \cdot \alpha}{\color{blue}{\beta \cdot \beta}} \]

       3. times-frac 26.1%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha}{\beta}} \]

   14. Simplified 26.1%

       \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 20: 70.8% accurate, 53.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 17.0%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
2. Step-by-step derivation

   1. associate-/l/ 15.6%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

   2. associate-*l* 15.5%

      \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. times-frac 21.5%

      \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

3. Simplified 41.7%

   \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

4. Taylor expanded in i around inf 64.6%

   \[\leadsto \color{blue}{0.0625} \]

5. Final simplification 64.6%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023283 
(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)))