Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 84.6%

Time: 13.9s

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: 16.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+206}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(\left(-i\right) - \alpha\right)}{-\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5e+154)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 2.95e+196)
     (* (/ i beta) (/ i beta))
     (if (<= beta 6.2e+206)
       0.0625
       (/ (* (/ i beta) (- (- i) alpha)) (- beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 2.95e+196) {
		tmp = (i / beta) * (i / beta);
	} else if (beta <= 6.2e+206) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (-i - alpha)) / -beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5d+154) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 2.95d+196) then
        tmp = (i / beta) * (i / beta)
    else if (beta <= 6.2d+206) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * (-i - alpha)) / -beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 2.95e+196) {
		tmp = (i / beta) * (i / beta);
	} else if (beta <= 6.2e+206) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (-i - alpha)) / -beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5e+154:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 2.95e+196:
		tmp = (i / beta) * (i / beta)
	elif beta <= 6.2e+206:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * (-i - alpha)) / -beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5e+154)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 2.95e+196)
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	elseif (beta <= 6.2e+206)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(Float64(-i) - alpha)) / Float64(-beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5e+154)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 2.95e+196)
		tmp = (i / beta) * (i / beta);
	elseif (beta <= 6.2e+206)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * (-i - alpha)) / -beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5e+154], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.95e+196], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.2e+206], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * N[((-i) - alpha), $MachinePrecision]), $MachinePrecision] / (-beta)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 5.00000000000000004e154

   1. Initial program 23.6%

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

   2. Taylor expanded in i around inf 41.4%

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

      1. *-commutative 41.4%

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

      2. unpow2 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in i around inf 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot {i}^{2}} - 1} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot \color{blue}{\left(i \cdot i\right)} - 1} \]

   7. Simplified 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot \left(i \cdot i\right)} - 1} \]

   8. Taylor expanded in i around inf 76.5%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 76.5%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]

      2. metadata-eval 76.5%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]

      3. unpow2 76.5%

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

   10. Simplified 76.5%

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

   if 5.00000000000000004e154 < beta < 2.9499999999999999e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 2.2%

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

   4. Taylor expanded in beta around inf 9.3%

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

      1. *-commutative 9.3%

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

      2. associate-/l* 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

   6. Simplified 18.0%

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

      1. associate-/r/ 18.0%

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

      2. +-commutative 18.0%

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

   8. Applied egg-rr 18.0%

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

      1. clear-num 18.0%

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

      2. inv-pow 18.0%

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

      3. add-sqr-sqrt 18.0%

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

      4. sqrt-unprod 2.2%

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

      5. sqr-neg 2.2%

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

      6. mul-1-neg 2.2%

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

      7. mul-1-neg 2.2%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 18.0%

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

      10. associate-/l* 2.8%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 30.7%

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

      13. mul-1-neg 30.7%

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

      14. mul-1-neg 30.7%

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

      15. sqr-neg 30.7%

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

      16. sqrt-unprod 38.9%

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

      17. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. unpow-1 38.9%

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

       2. associate-/r/ 42.0%

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

   12. Simplified 42.0%

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

   13. Taylor expanded in i around inf 16.4%

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

       1. unpow2 16.4%

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

       2. associate-/r* 43.0%

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

       3. unpow2 43.0%

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

       4. associate-*l/ 50.7%

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

       5. associate-*r/ 51.0%

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

   15. Simplified 51.0%

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

   if 2.9499999999999999e196 < beta < 6.19999999999999981e206

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 64.8%

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

   if 6.19999999999999981e206 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in beta around inf 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/l* 33.3%

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

      3. +-commutative 33.3%

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

      4. unpow2 33.3%

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

   6. Simplified 33.3%

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

      1. associate-/r/ 33.3%

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

      2. +-commutative 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

      3. associate-*l/ 30.7%

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

      4. sqrt-div 30.7%

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

      5. sqrt-prod 40.0%

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

      6. add-sqr-sqrt 40.0%

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

   10. Applied egg-rr 40.0%

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

       1. unpow2 40.0%

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

       2. clear-num 40.0%

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

       3. frac-2neg 40.0%

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

       4. frac-times 32.2%

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

       5. *-un-lft-identity 32.2%

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

       6. *-commutative 32.2%

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

       7. *-commutative 32.2%

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

   12. Applied egg-rr 32.2%

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

       1. *-commutative 32.2%

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

       2. associate-*r/ 30.7%

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

       3. associate-/l* 30.7%

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

       4. *-commutative 30.7%

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

       5. *-commutative 30.7%

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

       6. associate-/r* 40.0%

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

       7. distribute-rgt-neg-out 40.0%

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

       8. rem-square-sqrt 40.0%

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

       9. distribute-rgt-neg-in 40.0%

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

       10. *-commutative 40.0%

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

       11. associate-*r/ 78.2%

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

       12. distribute-lft-neg-out 78.2%

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

       13. distribute-rgt-neg-in 78.2%

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

       14. distribute-neg-frac 78.2%

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

   14. Simplified 78.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+206}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(\left(-i\right) - \alpha\right)}{-\beta}\\ \end{array} \]

Alternative 2: 84.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+207}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i + \alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.6e+154)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (<= beta 2.95e+196)
     (* (/ i beta) (/ i beta))
     (if (<= beta 3.7e+207)
       0.0625
       (/ 1.0 (* (/ beta i) (/ beta (+ i alpha))))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.6e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 2.95e+196) {
		tmp = (i / beta) * (i / beta);
	} else if (beta <= 3.7e+207) {
		tmp = 0.0625;
	} else {
		tmp = 1.0 / ((beta / i) * (beta / (i + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.6d+154) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if (beta <= 2.95d+196) then
        tmp = (i / beta) * (i / beta)
    else if (beta <= 3.7d+207) then
        tmp = 0.0625d0
    else
        tmp = 1.0d0 / ((beta / i) * (beta / (i + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.6e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 2.95e+196) {
		tmp = (i / beta) * (i / beta);
	} else if (beta <= 3.7e+207) {
		tmp = 0.0625;
	} else {
		tmp = 1.0 / ((beta / i) * (beta / (i + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.6e+154:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif beta <= 2.95e+196:
		tmp = (i / beta) * (i / beta)
	elif beta <= 3.7e+207:
		tmp = 0.0625
	else:
		tmp = 1.0 / ((beta / i) * (beta / (i + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.6e+154)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 2.95e+196)
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	elseif (beta <= 3.7e+207)
		tmp = 0.0625;
	else
		tmp = Float64(1.0 / Float64(Float64(beta / i) * Float64(beta / Float64(i + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.6e+154)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif (beta <= 2.95e+196)
		tmp = (i / beta) * (i / beta);
	elseif (beta <= 3.7e+207)
		tmp = 0.0625;
	else
		tmp = 1.0 / ((beta / i) * (beta / (i + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.6e+154], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.95e+196], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.7e+207], 0.0625, N[(1.0 / N[(N[(beta / i), $MachinePrecision] * N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 5.5999999999999998e154

   1. Initial program 23.6%

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

   2. Taylor expanded in i around inf 41.4%

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

      1. *-commutative 41.4%

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

      2. unpow2 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in i around inf 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot {i}^{2}} - 1} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot \color{blue}{\left(i \cdot i\right)} - 1} \]

   7. Simplified 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot \left(i \cdot i\right)} - 1} \]

   8. Taylor expanded in i around inf 76.5%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 76.5%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]

      2. metadata-eval 76.5%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]

      3. unpow2 76.5%

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

   10. Simplified 76.5%

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

   if 5.5999999999999998e154 < beta < 2.9499999999999999e196

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 2.2%

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

   4. Taylor expanded in beta around inf 9.3%

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

      1. *-commutative 9.3%

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

      2. associate-/l* 18.0%

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

      3. +-commutative 18.0%

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

      4. unpow2 18.0%

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

   6. Simplified 18.0%

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

      1. associate-/r/ 18.0%

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

      2. +-commutative 18.0%

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

   8. Applied egg-rr 18.0%

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

      1. clear-num 18.0%

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

      2. inv-pow 18.0%

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

      3. add-sqr-sqrt 18.0%

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

      4. sqrt-unprod 2.2%

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

      5. sqr-neg 2.2%

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

      6. mul-1-neg 2.2%

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

      7. mul-1-neg 2.2%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 18.0%

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

      10. associate-/l* 2.8%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 30.7%

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

      13. mul-1-neg 30.7%

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

      14. mul-1-neg 30.7%

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

      15. sqr-neg 30.7%

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

      16. sqrt-unprod 38.9%

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

      17. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. unpow-1 38.9%

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

       2. associate-/r/ 42.0%

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

   12. Simplified 42.0%

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

   13. Taylor expanded in i around inf 16.4%

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

       1. unpow2 16.4%

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

       2. associate-/r* 43.0%

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

       3. unpow2 43.0%

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

       4. associate-*l/ 50.7%

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

       5. associate-*r/ 51.0%

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

   15. Simplified 51.0%

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

   if 2.9499999999999999e196 < beta < 3.7e207

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 64.8%

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

   if 3.7e207 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in beta around inf 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/l* 33.3%

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

      3. +-commutative 33.3%

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

      4. unpow2 33.3%

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

   6. Simplified 33.3%

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

      1. associate-/r/ 33.3%

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

      2. +-commutative 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

      3. associate-*l/ 30.7%

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

      4. sqrt-div 30.7%

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

      5. sqrt-prod 40.0%

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

      6. add-sqr-sqrt 40.0%

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

   10. Applied egg-rr 40.0%

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

       1. unpow2 40.0%

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

       2. clear-num 40.0%

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

       3. clear-num 39.9%

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

       4. frac-times 35.8%

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

       5. metadata-eval 35.8%

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

       6. times-frac 30.7%

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

       7. add-sqr-sqrt 30.7%

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

       8. *-commutative 30.7%

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

       9. frac-times 73.9%

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

   12. Applied egg-rr 73.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i + \alpha}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+207}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{i} \cdot \frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 3: 83.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+196} \lor \neg \left(\beta \leq 3.1 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.1e+154)
   0.0625
   (if (or (<= beta 2.9e+196) (not (<= beta 3.1e+206)))
     (* (/ i beta) (/ i beta))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.1e+154) {
		tmp = 0.0625;
	} else if ((beta <= 2.9e+196) || !(beta <= 3.1e+206)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.1d+154) then
        tmp = 0.0625d0
    else if ((beta <= 2.9d+196) .or. (.not. (beta <= 3.1d+206))) then
        tmp = (i / beta) * (i / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.1e+154) {
		tmp = 0.0625;
	} else if ((beta <= 2.9e+196) || !(beta <= 3.1e+206)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.1e+154:
		tmp = 0.0625
	elif (beta <= 2.9e+196) or not (beta <= 3.1e+206):
		tmp = (i / beta) * (i / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.1e+154)
		tmp = 0.0625;
	elseif ((beta <= 2.9e+196) || !(beta <= 3.1e+206))
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.1e+154)
		tmp = 0.0625;
	elseif ((beta <= 2.9e+196) || ~((beta <= 3.1e+206)))
		tmp = (i / beta) * (i / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.1e+154], 0.0625, If[Or[LessEqual[beta, 2.9e+196], N[Not[LessEqual[beta, 3.1e+206]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0999999999999999e154 or 2.9e196 < beta < 3.09999999999999991e206

   1. Initial program 23.2%

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

      1. associate-/l/ 20.9%

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

      2. associate-*l* 20.8%

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

      3. times-frac 30.3%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in i around inf 76.0%

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

   if 5.0999999999999999e154 < beta < 2.9e196 or 3.09999999999999991e206 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in beta around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 28.0%

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

      3. +-commutative 28.0%

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

      4. unpow2 28.0%

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

   6. Simplified 28.0%

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

      1. associate-/r/ 28.0%

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

      2. +-commutative 28.0%

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

   8. Applied egg-rr 28.0%

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

      1. clear-num 28.0%

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

      2. inv-pow 28.0%

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

      3. add-sqr-sqrt 28.0%

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

      4. sqrt-unprod 20.7%

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

      5. sqr-neg 20.7%

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

      6. mul-1-neg 20.7%

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

      7. mul-1-neg 20.7%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 28.0%

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

      10. associate-/l* 22.3%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 31.7%

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

      13. mul-1-neg 31.7%

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

      14. mul-1-neg 31.7%

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

      15. sqr-neg 31.7%

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

      16. sqrt-unprod 40.5%

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

      17. add-sqr-sqrt 40.5%

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

   10. Applied egg-rr 40.5%

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

       1. unpow-1 40.5%

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

       2. associate-/r/ 44.5%

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

   12. Simplified 44.5%

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

   13. Taylor expanded in i around inf 25.7%

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

       1. unpow2 25.7%

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

       2. associate-/r* 40.0%

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

       3. unpow2 40.0%

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

       4. associate-*l/ 65.3%

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

       5. associate-*r/ 65.4%

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

   15. Simplified 65.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+196} \lor \neg \left(\beta \leq 3.1 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4: 83.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196} \lor \neg \left(\beta \leq 1.1 \cdot 10^{+207}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.5e+154)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (if (or (<= beta 2.95e+196) (not (<= beta 1.1e+207)))
     (* (/ i beta) (/ i beta))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if ((beta <= 2.95e+196) || !(beta <= 1.1e+207)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.5d+154) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else if ((beta <= 2.95d+196) .or. (.not. (beta <= 1.1d+207))) then
        tmp = (i / beta) * (i / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+154) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if ((beta <= 2.95e+196) || !(beta <= 1.1e+207)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.5e+154:
		tmp = 0.0625 + (0.015625 / (i * i))
	elif (beta <= 2.95e+196) or not (beta <= 1.1e+207):
		tmp = (i / beta) * (i / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+154)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif ((beta <= 2.95e+196) || !(beta <= 1.1e+207))
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.5e+154)
		tmp = 0.0625 + (0.015625 / (i * i));
	elseif ((beta <= 2.95e+196) || ~((beta <= 1.1e+207)))
		tmp = (i / beta) * (i / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+154], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[beta, 2.95e+196], N[Not[LessEqual[beta, 1.1e+207]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.5000000000000006e154

   1. Initial program 23.6%

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

   2. Taylor expanded in i around inf 41.4%

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

      1. *-commutative 41.4%

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

      2. unpow2 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in i around inf 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot {i}^{2}} - 1} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{4 \cdot \color{blue}{\left(i \cdot i\right)} - 1} \]

   7. Simplified 38.0%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot \left(i \cdot i\right)} - 1} \]

   8. Taylor expanded in i around inf 76.5%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 76.5%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]

      2. metadata-eval 76.5%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]

      3. unpow2 76.5%

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

   10. Simplified 76.5%

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

   if 5.5000000000000006e154 < beta < 2.9499999999999999e196 or 1.10000000000000004e207 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in beta around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 28.0%

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

      3. +-commutative 28.0%

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

      4. unpow2 28.0%

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

   6. Simplified 28.0%

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

      1. associate-/r/ 28.0%

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

      2. +-commutative 28.0%

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

   8. Applied egg-rr 28.0%

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

      1. clear-num 28.0%

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

      2. inv-pow 28.0%

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

      3. add-sqr-sqrt 28.0%

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

      4. sqrt-unprod 20.7%

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

      5. sqr-neg 20.7%

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

      6. mul-1-neg 20.7%

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

      7. mul-1-neg 20.7%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 28.0%

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

      10. associate-/l* 22.3%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 31.7%

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

      13. mul-1-neg 31.7%

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

      14. mul-1-neg 31.7%

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

      15. sqr-neg 31.7%

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

      16. sqrt-unprod 40.5%

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

      17. add-sqr-sqrt 40.5%

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

   10. Applied egg-rr 40.5%

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

       1. unpow-1 40.5%

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

       2. associate-/r/ 44.5%

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

   12. Simplified 44.5%

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

   13. Taylor expanded in i around inf 25.7%

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

       1. unpow2 25.7%

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

       2. associate-/r* 40.0%

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

       3. unpow2 40.0%

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

       4. associate-*l/ 65.3%

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

       5. associate-*r/ 65.4%

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

   15. Simplified 65.4%

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

   if 2.9499999999999999e196 < beta < 1.10000000000000004e207

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 64.8%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+154}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+196} \lor \neg \left(\beta \leq 1.1 \cdot 10^{+207}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5: 75.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+215}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.6e+215) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.6e+215) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.6d+215) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.6e+215) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.6e+215:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.6e+215)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.6e+215)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.6e+215], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5999999999999999e215

   1. Initial program 21.7%

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

      1. associate-/l/ 19.5%

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

      2. associate-*l* 19.5%

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

      3. times-frac 28.3%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in i around inf 74.3%

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

   if 1.5999999999999999e215 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in beta around inf 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-/l* 34.5%

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

      3. +-commutative 34.5%

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

      4. unpow2 34.5%

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

   6. Simplified 34.5%

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

   7. Taylor expanded in alpha around inf 33.4%

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

      1. unpow2 33.4%

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

   9. Simplified 33.4%

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

       1. times-frac 40.6%

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

   11. Applied egg-rr 40.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+215}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 6: 71.1% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.6%

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

   1. associate-/l/ 17.6%

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

   2. associate-*l* 17.6%

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

   3. times-frac 25.6%

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

3. Simplified 38.9%

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

4. Taylor expanded in i around inf 68.5%

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

5. Final simplification 68.5%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023268 
(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)))