Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 84.4%

Time: 20.7s

Alternatives: 6

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: 15.7% 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, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625 + \left(0.0625 \cdot \frac{\beta \cdot \beta}{i \cdot i} - 0.00390625 \cdot \frac{4 \cdot \left(\beta \cdot \beta + -1\right) + \left(\beta \cdot \beta\right) \cdot 20}{i \cdot i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+111)
   (+
    0.0625
    (-
     (* 0.0625 (/ (* beta beta) (* i i)))
     (*
      0.00390625
      (/ (+ (* 4.0 (+ (* beta beta) -1.0)) (* (* beta beta) 20.0)) (* i i)))))
   (* (/ i (+ (+ beta alpha) (* i 2.0))) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+111) {
		tmp = 0.0625 + ((0.0625 * ((beta * beta) / (i * i))) - (0.00390625 * (((4.0 * ((beta * beta) + -1.0)) + ((beta * beta) * 20.0)) / (i * i))));
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.5d+111) then
        tmp = 0.0625d0 + ((0.0625d0 * ((beta * beta) / (i * i))) - (0.00390625d0 * (((4.0d0 * ((beta * beta) + (-1.0d0))) + ((beta * beta) * 20.0d0)) / (i * i))))
    else
        tmp = (i / ((beta + alpha) + (i * 2.0d0))) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+111) {
		tmp = 0.0625 + ((0.0625 * ((beta * beta) / (i * i))) - (0.00390625 * (((4.0 * ((beta * beta) + -1.0)) + ((beta * beta) * 20.0)) / (i * i))));
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+111:
		tmp = 0.0625 + ((0.0625 * ((beta * beta) / (i * i))) - (0.00390625 * (((4.0 * ((beta * beta) + -1.0)) + ((beta * beta) * 20.0)) / (i * i))))
	else:
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+111)
		tmp = Float64(0.0625 + Float64(Float64(0.0625 * Float64(Float64(beta * beta) / Float64(i * i))) - Float64(0.00390625 * Float64(Float64(Float64(4.0 * Float64(Float64(beta * beta) + -1.0)) + Float64(Float64(beta * beta) * 20.0)) / Float64(i * i)))));
	else
		tmp = Float64(Float64(i / Float64(Float64(beta + alpha) + Float64(i * 2.0))) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+111)
		tmp = 0.0625 + ((0.0625 * ((beta * beta) / (i * i))) - (0.00390625 * (((4.0 * ((beta * beta) + -1.0)) + ((beta * beta) * 20.0)) / (i * i))));
	else
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.5e+111], N[(0.0625 + N[(N[(0.0625 * N[(N[(beta * beta), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.00390625 * N[(N[(N[(4.0 * N[(N[(beta * beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.49999999999999948e111

   1. Initial program 20.3%

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

      1. associate-/l/ 18.1%

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

      2. associate-/l* 21.6%

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

      3. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   5. Taylor expanded in alpha around 0 17.8%

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

      1. times-frac 43.3%

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

      2. unpow2 43.3%

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

      3. unpow2 43.3%

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

      4. unpow2 43.3%

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

      5. sub-neg 43.3%

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

      6. unpow2 43.3%

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

      7. metadata-eval 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in i around inf 82.8%

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

      1. associate--l+ 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. sub-neg 82.8%

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

      5. unpow2 82.8%

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

      6. metadata-eval 82.8%

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

      7. distribute-rgt-out 82.8%

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

      8. unpow2 82.8%

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

      9. metadata-eval 82.8%

         \[\leadsto 0.0625 + \left(0.0625 \cdot \frac{\beta \cdot \beta}{i \cdot i} - 0.00390625 \cdot \frac{4 \cdot \left(\beta \cdot \beta + -1\right) + \left(\beta \cdot \beta\right) \cdot \color{blue}{20}}{{i}^{2}}\right) \]

      10. unpow2 82.8%

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

   10. Simplified 82.8%

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

   if 7.49999999999999948e111 < beta

   1. Initial program 2.5%

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

      1. associate-/l/ 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* 3.6%

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

      3. +-commutative 3.6%

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

   3. Simplified 3.6%

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

      1. associate-*r/ 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. +-commutative 0.1%

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

      5. *-commutative 0.1%

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

      6. +-commutative 0.1%

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

      7. associate-*l* 0.1%

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

      8. associate-*l* 0.1%

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

      9. times-frac 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-+l+ 0.1%

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

      3. *-commutative 0.1%

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

      4. associate-/l* 3.6%

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

      5. +-commutative 3.6%

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

      6. *-commutative 3.6%

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

   8. Simplified 3.6%

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

   9. Taylor expanded in beta around inf 57.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625 + \left(0.0625 \cdot \frac{\beta \cdot \beta}{i \cdot i} - 0.00390625 \cdot \frac{4 \cdot \left(\beta \cdot \beta + -1\right) + \left(\beta \cdot \beta\right) \cdot 20}{i \cdot i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \frac{\beta + i \cdot 2}{i}\\ \mathbf{if}\;\beta \leq 10^{+117}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0} \cdot \left(0.25 - \frac{\beta \cdot -0.25}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ beta (* i 2.0)) i)))
   (if (<= beta 1e+117)
     (* (/ 1.0 (* t_0 t_0)) (- 0.25 (/ (* beta -0.25) i)))
     (* (/ i (+ (+ beta alpha) (* i 2.0))) (/ (+ i alpha) beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + (i * 2.0)) / i;
	double tmp;
	if (beta <= 1e+117) {
		tmp = (1.0 / (t_0 * t_0)) * (0.25 - ((beta * -0.25) / i));
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + (i * 2.0d0)) / i
    if (beta <= 1d+117) then
        tmp = (1.0d0 / (t_0 * t_0)) * (0.25d0 - ((beta * (-0.25d0)) / i))
    else
        tmp = (i / ((beta + alpha) + (i * 2.0d0))) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + (i * 2.0)) / i;
	double tmp;
	if (beta <= 1e+117) {
		tmp = (1.0 / (t_0 * t_0)) * (0.25 - ((beta * -0.25) / i));
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + (i * 2.0)) / i
	tmp = 0
	if beta <= 1e+117:
		tmp = (1.0 / (t_0 * t_0)) * (0.25 - ((beta * -0.25) / i))
	else:
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + Float64(i * 2.0)) / i)
	tmp = 0.0
	if (beta <= 1e+117)
		tmp = Float64(Float64(1.0 / Float64(t_0 * t_0)) * Float64(0.25 - Float64(Float64(beta * -0.25) / i)));
	else
		tmp = Float64(Float64(i / Float64(Float64(beta + alpha) + Float64(i * 2.0))) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + (i * 2.0)) / i;
	tmp = 0.0;
	if (beta <= 1e+117)
		tmp = (1.0 / (t_0 * t_0)) * (0.25 - ((beta * -0.25) / i));
	else
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[beta, 1e+117], N[(N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.25 - N[(N[(beta * -0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000005e117

   1. Initial program 20.0%

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

      1. associate-/l/ 17.9%

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

      2. associate-/l* 21.3%

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

      3. +-commutative 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in alpha around 0 17.6%

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

      1. times-frac 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. unpow2 42.7%

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

      5. sub-neg 42.7%

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

      6. unpow2 42.7%

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

      7. metadata-eval 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in i around -inf 41.1%

      \[\leadsto \frac{i \cdot i}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)} \cdot \color{blue}{\left(0.25 + -1 \cdot \frac{-0.5 \cdot \beta - -0.25 \cdot \beta}{i}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-out-- 41.1%

         \[\leadsto \frac{i \cdot i}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)} \cdot \left(0.25 + -1 \cdot \frac{\color{blue}{\beta \cdot \left(-0.5 - -0.25\right)}}{i}\right) \]

      2. metadata-eval 41.1%

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

   10. Simplified 41.1%

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

       1. clear-num 41.1%

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

   12. Applied egg-rr 41.1%

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

       1. times-frac 82.2%

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

   14. Simplified 82.2%

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

   if 1.00000000000000005e117 < beta

   1. Initial program 2.6%

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

      1. associate-/l/ 0.1%

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

      2. associate-/l* 3.9%

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

      3. +-commutative 3.9%

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

   3. Simplified 3.9%

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

      1. associate-*r/ 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. +-commutative 0.1%

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

      5. *-commutative 0.1%

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

      6. +-commutative 0.1%

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

      7. associate-*l* 0.1%

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

      8. associate-*l* 0.1%

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

      9. times-frac 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-+l+ 0.1%

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

      3. *-commutative 0.1%

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

      4. associate-/l* 3.9%

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

      5. +-commutative 3.9%

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

      6. *-commutative 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in beta around inf 61.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+117}:\\ \;\;\;\;\frac{1}{\frac{\beta + i \cdot 2}{i} \cdot \frac{\beta + i \cdot 2}{i}} \cdot \left(0.25 - \frac{\beta \cdot -0.25}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \frac{i}{\beta + i \cdot 2}\\ \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+115}:\\ \;\;\;\;t\_0 \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta}{i}\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ i (+ beta (* i 2.0)))))
   (if (<= beta 5.7e+115)
     (* t_0 (* (+ 0.25 (* 0.25 (/ beta i))) t_0))
     (* (/ i (+ (+ beta alpha) (* i 2.0))) (/ (+ i alpha) beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = i / (beta + (i * 2.0));
	double tmp;
	if (beta <= 5.7e+115) {
		tmp = t_0 * ((0.25 + (0.25 * (beta / i))) * t_0);
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = i / (beta + (i * 2.0d0))
    if (beta <= 5.7d+115) then
        tmp = t_0 * ((0.25d0 + (0.25d0 * (beta / i))) * t_0)
    else
        tmp = (i / ((beta + alpha) + (i * 2.0d0))) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = i / (beta + (i * 2.0));
	double tmp;
	if (beta <= 5.7e+115) {
		tmp = t_0 * ((0.25 + (0.25 * (beta / i))) * t_0);
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = i / (beta + (i * 2.0))
	tmp = 0
	if beta <= 5.7e+115:
		tmp = t_0 * ((0.25 + (0.25 * (beta / i))) * t_0)
	else:
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(i / Float64(beta + Float64(i * 2.0)))
	tmp = 0.0
	if (beta <= 5.7e+115)
		tmp = Float64(t_0 * Float64(Float64(0.25 + Float64(0.25 * Float64(beta / i))) * t_0));
	else
		tmp = Float64(Float64(i / Float64(Float64(beta + alpha) + Float64(i * 2.0))) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = i / (beta + (i * 2.0));
	tmp = 0.0;
	if (beta <= 5.7e+115)
		tmp = t_0 * ((0.25 + (0.25 * (beta / i))) * t_0);
	else
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.7e+115], N[(t$95$0 * N[(N[(0.25 + N[(0.25 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.69999999999999965e115

   1. Initial program 20.1%

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

      1. associate-/l/ 17.9%

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

      2. associate-/l* 21.4%

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

      3. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around 0 17.7%

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

      1. times-frac 42.9%

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

      2. unpow2 42.9%

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

      3. unpow2 42.9%

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

      4. unpow2 42.9%

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

      5. sub-neg 42.9%

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

      6. unpow2 42.9%

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

      7. metadata-eval 42.9%

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

   7. Simplified 42.9%

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

   8. Taylor expanded in i around -inf 41.3%

      \[\leadsto \frac{i \cdot i}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)} \cdot \color{blue}{\left(0.25 + -1 \cdot \frac{-0.5 \cdot \beta - -0.25 \cdot \beta}{i}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-out-- 41.3%

         \[\leadsto \frac{i \cdot i}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)} \cdot \left(0.25 + -1 \cdot \frac{\color{blue}{\beta \cdot \left(-0.5 - -0.25\right)}}{i}\right) \]

      2. metadata-eval 41.3%

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

   10. Simplified 41.3%

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

       1. distribute-lft-in 41.3%

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

       2. times-frac 41.3%

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

       3. *-commutative 41.3%

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

       4. *-commutative 41.3%

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

       5. times-frac 82.5%

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

       6. *-commutative 82.5%

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

       7. *-commutative 82.5%

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

       8. associate-/l* 82.5%

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

   12. Applied egg-rr 82.5%

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

       1. distribute-lft-in 82.5%

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

       2. *-commutative 82.5%

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

       3. associate-*r* 82.5%

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

       4. mul-1-neg 82.5%

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

       5. associate-*r/ 82.5%

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

       6. *-commutative 82.5%

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

       7. associate-*r/ 82.5%

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

       8. distribute-lft-neg-in 82.5%

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

       9. metadata-eval 82.5%

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

       10. *-commutative 82.5%

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

   14. Simplified 82.5%

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

   if 5.69999999999999965e115 < beta

   1. Initial program 2.6%

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

      1. associate-/l/ 0.1%

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

      2. associate-/l* 3.8%

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

      3. +-commutative 3.8%

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

   3. Simplified 3.8%

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

      1. associate-*r/ 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. +-commutative 0.1%

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

      5. *-commutative 0.1%

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

      6. +-commutative 0.1%

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

      7. associate-*l* 0.1%

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

      8. associate-*l* 0.1%

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

      9. times-frac 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-+l+ 0.1%

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

      3. *-commutative 0.1%

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

      4. associate-/l* 3.8%

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

      5. +-commutative 3.8%

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

      6. *-commutative 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in beta around inf 60.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{i}{\beta + i \cdot 2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta}{i}\right) \cdot \frac{i}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+111)
   0.0625
   (* (/ i (+ (+ beta alpha) (* i 2.0))) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+111) {
		tmp = 0.0625;
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.5d+111) then
        tmp = 0.0625d0
    else
        tmp = (i / ((beta + alpha) + (i * 2.0d0))) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+111) {
		tmp = 0.0625;
	} else {
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+111:
		tmp = 0.0625
	else:
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+111)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(Float64(beta + alpha) + Float64(i * 2.0))) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+111)
		tmp = 0.0625;
	else
		tmp = (i / ((beta + alpha) + (i * 2.0))) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.5e+111], 0.0625, N[(N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.49999999999999948e111

   1. Initial program 20.3%

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

      1. associate-/l/ 18.1%

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

      2. associate-/l* 21.6%

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

      3. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   5. Taylor expanded in i around inf 82.7%

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

   if 7.49999999999999948e111 < beta

   1. Initial program 2.5%

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

      1. associate-/l/ 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* 3.6%

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

      3. +-commutative 3.6%

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

   3. Simplified 3.6%

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

      1. associate-*r/ 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. +-commutative 0.1%

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

      5. *-commutative 0.1%

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

      6. +-commutative 0.1%

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

      7. associate-*l* 0.1%

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

      8. associate-*l* 0.1%

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

      9. times-frac 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-+l+ 0.1%

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

      3. *-commutative 0.1%

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

      4. associate-/l* 3.6%

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

      5. +-commutative 3.6%

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

      6. *-commutative 3.6%

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

   8. Simplified 3.6%

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

   9. Taylor expanded in beta around inf 57.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+116}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.8e+116) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+116) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.8d+116) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.8e+116) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.8e+116:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.8e+116)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.8e+116)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.8e+116], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.8000000000000003e116

   1. Initial program 20.0%

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

      1. associate-/l/ 17.9%

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

      2. associate-/l* 21.3%

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

      3. +-commutative 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in i around inf 82.2%

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

   if 5.8000000000000003e116 < beta

   1. Initial program 2.6%

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

      1. associate-/l/ 0.1%

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

      2. associate-/l* 3.9%

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

      3. +-commutative 3.9%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in alpha around 0 0.1%

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

      1. times-frac 16.3%

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

      2. unpow2 16.3%

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

      3. unpow2 16.3%

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

      4. unpow2 16.3%

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

      5. sub-neg 16.3%

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

      6. unpow2 16.3%

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

      7. metadata-eval 16.3%

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

   7. Simplified 16.3%

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

   8. Taylor expanded in beta around -inf 32.1%

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

      1. distribute-rgt-out-- 32.1%

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

      2. metadata-eval 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in i around 0 32.4%

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

       1. unpow2 32.4%

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

       2. unpow2 32.4%

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

       3. times-frac 56.7%

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

   13. Simplified 56.7%

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

Alternative 6: 71.8% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

   1. associate-/l/ 14.9%

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

   2. associate-/l* 18.3%

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

   3. +-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in i around inf 72.9%

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

Reproduce

herbie shell --seed 2024098 
(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)))