Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 85.1%

Time: 19.1s

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: 85.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.0625 \cdot \left(2 \cdot \alpha + \beta \cdot 2\right) - 0.125 \cdot \left(\beta + \alpha\right)\\ t_1 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+120}:\\ \;\;\;\;0.0625 + \frac{\left(-2 \cdot \frac{\left(\beta + \alpha\right) \cdot t\_0}{i} + -0.00390625 \cdot \frac{4 \cdot \left(-1 + t\_1\right) + \left(t\_1 \cdot 4 + t\_1 \cdot 16\right)}{i}\right) + \left(t\_0 + 0.0625 \cdot \frac{\beta \cdot \alpha + t\_1}{i}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\alpha + i}}{\beta} \cdot \sqrt{i}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0
         (-
          (* 0.0625 (+ (* 2.0 alpha) (* beta 2.0)))
          (* 0.125 (+ beta alpha))))
        (t_1 (pow (+ beta alpha) 2.0)))
   (if (<= beta 2.2e+120)
     (+
      0.0625
      (/
       (+
        (+
         (* -2.0 (/ (* (+ beta alpha) t_0) i))
         (*
          -0.00390625
          (/ (+ (* 4.0 (+ -1.0 t_1)) (+ (* t_1 4.0) (* t_1 16.0))) i)))
        (+ t_0 (* 0.0625 (/ (+ (* beta alpha) t_1) i))))
       i))
     (pow (* (/ (sqrt (+ alpha i)) beta) (sqrt i)) 2.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (0.0625 * ((2.0 * alpha) + (beta * 2.0))) - (0.125 * (beta + alpha));
	double t_1 = pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 2.2e+120) {
		tmp = 0.0625 + ((((-2.0 * (((beta + alpha) * t_0) / i)) + (-0.00390625 * (((4.0 * (-1.0 + t_1)) + ((t_1 * 4.0) + (t_1 * 16.0))) / i))) + (t_0 + (0.0625 * (((beta * alpha) + t_1) / i)))) / i);
	} else {
		tmp = pow(((sqrt((alpha + i)) / beta) * sqrt(i)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.0625d0 * ((2.0d0 * alpha) + (beta * 2.0d0))) - (0.125d0 * (beta + alpha))
    t_1 = (beta + alpha) ** 2.0d0
    if (beta <= 2.2d+120) then
        tmp = 0.0625d0 + (((((-2.0d0) * (((beta + alpha) * t_0) / i)) + ((-0.00390625d0) * (((4.0d0 * ((-1.0d0) + t_1)) + ((t_1 * 4.0d0) + (t_1 * 16.0d0))) / i))) + (t_0 + (0.0625d0 * (((beta * alpha) + t_1) / i)))) / i)
    else
        tmp = ((sqrt((alpha + i)) / beta) * sqrt(i)) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (0.0625 * ((2.0 * alpha) + (beta * 2.0))) - (0.125 * (beta + alpha));
	double t_1 = Math.pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 2.2e+120) {
		tmp = 0.0625 + ((((-2.0 * (((beta + alpha) * t_0) / i)) + (-0.00390625 * (((4.0 * (-1.0 + t_1)) + ((t_1 * 4.0) + (t_1 * 16.0))) / i))) + (t_0 + (0.0625 * (((beta * alpha) + t_1) / i)))) / i);
	} else {
		tmp = Math.pow(((Math.sqrt((alpha + i)) / beta) * Math.sqrt(i)), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (0.0625 * ((2.0 * alpha) + (beta * 2.0))) - (0.125 * (beta + alpha))
	t_1 = math.pow((beta + alpha), 2.0)
	tmp = 0
	if beta <= 2.2e+120:
		tmp = 0.0625 + ((((-2.0 * (((beta + alpha) * t_0) / i)) + (-0.00390625 * (((4.0 * (-1.0 + t_1)) + ((t_1 * 4.0) + (t_1 * 16.0))) / i))) + (t_0 + (0.0625 * (((beta * alpha) + t_1) / i)))) / i)
	else:
		tmp = math.pow(((math.sqrt((alpha + i)) / beta) * math.sqrt(i)), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(0.0625 * Float64(Float64(2.0 * alpha) + Float64(beta * 2.0))) - Float64(0.125 * Float64(beta + alpha)))
	t_1 = Float64(beta + alpha) ^ 2.0
	tmp = 0.0
	if (beta <= 2.2e+120)
		tmp = Float64(0.0625 + Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(Float64(beta + alpha) * t_0) / i)) + Float64(-0.00390625 * Float64(Float64(Float64(4.0 * Float64(-1.0 + t_1)) + Float64(Float64(t_1 * 4.0) + Float64(t_1 * 16.0))) / i))) + Float64(t_0 + Float64(0.0625 * Float64(Float64(Float64(beta * alpha) + t_1) / i)))) / i));
	else
		tmp = Float64(Float64(sqrt(Float64(alpha + i)) / beta) * sqrt(i)) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (0.0625 * ((2.0 * alpha) + (beta * 2.0))) - (0.125 * (beta + alpha));
	t_1 = (beta + alpha) ^ 2.0;
	tmp = 0.0;
	if (beta <= 2.2e+120)
		tmp = 0.0625 + ((((-2.0 * (((beta + alpha) * t_0) / i)) + (-0.00390625 * (((4.0 * (-1.0 + t_1)) + ((t_1 * 4.0) + (t_1 * 16.0))) / i))) + (t_0 + (0.0625 * (((beta * alpha) + t_1) / i)))) / i);
	else
		tmp = ((sqrt((alpha + i)) / beta) * sqrt(i)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(0.0625 * N[(N[(2.0 * alpha), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 2.2e+120], N[(0.0625 + N[(N[(N[(N[(-2.0 * N[(N[(N[(beta + alpha), $MachinePrecision] * t$95$0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(-0.00390625 * N[(N[(N[(4.0 * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 4.0), $MachinePrecision] + N[(t$95$1 * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(0.0625 * N[(N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Sqrt[N[(alpha + i), $MachinePrecision]], $MachinePrecision] / beta), $MachinePrecision] * N[Sqrt[i], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000001e120

   1. Initial program 23.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in i around -inf 75.5%

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

   if 2.2000000000000001e120 < beta

   1. Initial program 0.3%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in beta around inf 35.8%

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

      1. add-exp-log 34.3%

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

      2. div-inv 34.3%

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

      3. pow-flip 34.3%

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

      4. metadata-eval 34.3%

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

   7. Applied egg-rr 34.3%

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

      1. rem-exp-log 34.3%

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

      2. add-sqr-sqrt 34.3%

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

      3. pow2 34.3%

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

      4. rem-exp-log 34.3%

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

      5. sqrt-prod 34.3%

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

      6. +-commutative 34.3%

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

      7. sqrt-pow1 48.7%

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

      8. metadata-eval 48.7%

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

      9. unpow-1 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. add-sqr-sqrt 48.7%

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

       2. pow2 48.7%

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

       3. *-commutative 48.7%

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

       4. sqrt-prod 48.7%

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

       5. rem-exp-log 51.0%

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

       6. sqrt-pow1 61.5%

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

       7. metadata-eval 61.5%

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

       8. pow1 61.5%

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

       9. un-div-inv 61.6%

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

   11. Applied egg-rr 61.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+120}:\\ \;\;\;\;0.0625 + \frac{\left(-2 \cdot \frac{\left(\beta + \alpha\right) \cdot \left(0.0625 \cdot \left(2 \cdot \alpha + \beta \cdot 2\right) - 0.125 \cdot \left(\beta + \alpha\right)\right)}{i} + -0.00390625 \cdot \frac{4 \cdot \left(-1 + {\left(\beta + \alpha\right)}^{2}\right) + \left({\left(\beta + \alpha\right)}^{2} \cdot 4 + {\left(\beta + \alpha\right)}^{2} \cdot 16\right)}{i}\right) + \left(\left(0.0625 \cdot \left(2 \cdot \alpha + \beta \cdot 2\right) - 0.125 \cdot \left(\beta + \alpha\right)\right) + 0.0625 \cdot \frac{\beta \cdot \alpha + {\left(\beta + \alpha\right)}^{2}}{i}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\alpha + i}}{\beta} \cdot \sqrt{i}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\alpha + i}}{\beta} \cdot \sqrt{i}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.5e+121)
   0.0625
   (pow (* (/ (sqrt (+ alpha i)) beta) (sqrt i)) 2.0)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+121) {
		tmp = 0.0625;
	} else {
		tmp = pow(((sqrt((alpha + i)) / beta) * sqrt(i)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.5d+121) then
        tmp = 0.0625d0
    else
        tmp = ((sqrt((alpha + i)) / beta) * sqrt(i)) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+121) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow(((Math.sqrt((alpha + i)) / beta) * Math.sqrt(i)), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.5e+121:
		tmp = 0.0625
	else:
		tmp = math.pow(((math.sqrt((alpha + i)) / beta) * math.sqrt(i)), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.5e+121)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(sqrt(Float64(alpha + i)) / beta) * sqrt(i)) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.5e+121)
		tmp = 0.0625;
	else
		tmp = ((sqrt((alpha + i)) / beta) * sqrt(i)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.5e+121], 0.0625, N[Power[N[(N[(N[Sqrt[N[(alpha + i), $MachinePrecision]], $MachinePrecision] / beta), $MachinePrecision] * N[Sqrt[i], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.50000000000000004e121

   1. Initial program 23.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in i around inf 81.7%

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

   if 2.50000000000000004e121 < beta

   1. Initial program 0.3%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in beta around inf 35.8%

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

      1. add-exp-log 34.3%

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

      2. div-inv 34.3%

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

      3. pow-flip 34.3%

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

      4. metadata-eval 34.3%

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

   7. Applied egg-rr 34.3%

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

      1. rem-exp-log 34.3%

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

      2. add-sqr-sqrt 34.3%

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

      3. pow2 34.3%

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

      4. rem-exp-log 34.3%

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

      5. sqrt-prod 34.3%

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

      6. +-commutative 34.3%

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

      7. sqrt-pow1 48.7%

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

      8. metadata-eval 48.7%

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

      9. unpow-1 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. add-sqr-sqrt 48.7%

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

       2. pow2 48.7%

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

       3. *-commutative 48.7%

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

       4. sqrt-prod 48.7%

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

       5. rem-exp-log 51.0%

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

       6. sqrt-pow1 61.5%

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

       7. metadata-eval 61.5%

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

       8. pow1 61.5%

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

       9. un-div-inv 61.6%

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

   11. Applied egg-rr 61.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\alpha + i}}{\beta} \cdot \sqrt{i}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := -1 + t\_1\\ t_3 := i \cdot \left(\left(\beta + \alpha\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(\beta + i\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (+ -1.0 t_1))
        (t_3 (* i (+ (+ beta alpha) i))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) INFINITY)
     (/
      (*
       (/ (* i (+ alpha (+ beta i))) (fma i 2.0 (+ beta alpha)))
       (/ (* i (+ beta i)) (+ beta (* 2.0 i))))
      t_2)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* 2.0 alpha) (* beta 2.0)) i)))
      (* 0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = -1.0 + t_1;
	double t_3 = i * ((beta + alpha) + i);
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (((i * (alpha + (beta + i))) / fma(i, 2.0, (beta + alpha))) * ((i * (beta + i)) / (beta + (2.0 * i)))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(-1.0 + t_1)
	t_3 = Float64(i * Float64(Float64(beta + alpha) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(Float64(i * Float64(alpha + Float64(beta + i))) / fma(i, 2.0, Float64(beta + alpha))) * Float64(Float64(i * Float64(beta + i)) / Float64(beta + Float64(2.0 * i)))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(2.0 * alpha) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 49.1%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. *-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-undefine 99.6%

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

      11. associate-+r+ 99.6%

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

      12. +-commutative 99.6%

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

      13. +-commutative 99.6%

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

      14. *-commutative 99.6%

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

      15. fma-undefine 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-+r+ 99.6%

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

      2. associate-+r+ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in alpha around 0 93.5%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.0%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in i around inf 71.5%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\beta + \alpha\right) + i\right)\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(\beta + i\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{1}{\beta} \cdot \left(\left(\alpha + i\right) \cdot \frac{1}{\beta}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+121)
   0.0625
   (* i (* (/ 1.0 beta) (* (+ alpha i) (/ 1.0 beta))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+121) {
		tmp = 0.0625;
	} else {
		tmp = i * ((1.0 / beta) * ((alpha + i) * (1.0 / beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1d+121) then
        tmp = 0.0625d0
    else
        tmp = i * ((1.0d0 / beta) * ((alpha + i) * (1.0d0 / beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+121) {
		tmp = 0.0625;
	} else {
		tmp = i * ((1.0 / beta) * ((alpha + i) * (1.0 / beta)));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1e+121:
		tmp = 0.0625
	else:
		tmp = i * ((1.0 / beta) * ((alpha + i) * (1.0 / beta)))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+121)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(1.0 / beta) * Float64(Float64(alpha + i) * Float64(1.0 / beta))));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1e+121)
		tmp = 0.0625;
	else
		tmp = i * ((1.0 / beta) * ((alpha + i) * (1.0 / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1e+121], 0.0625, N[(i * N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000004e121

   1. Initial program 23.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in i around inf 81.7%

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

   if 1.00000000000000004e121 < beta

   1. Initial program 0.3%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in beta around inf 35.8%

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

      1. add-exp-log 34.3%

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

      2. div-inv 34.3%

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

      3. pow-flip 34.3%

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

      4. metadata-eval 34.3%

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

   7. Applied egg-rr 34.3%

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

      1. rem-exp-log 35.7%

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

      2. *-commutative 35.7%

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

      3. sqr-pow 35.6%

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

      4. associate-*l* 51.3%

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

      5. metadata-eval 51.3%

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

      6. unpow-1 51.3%

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

      7. metadata-eval 51.3%

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

      8. unpow-1 51.3%

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

      9. +-commutative 51.3%

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

   9. Applied egg-rr 51.3%

      \[\leadsto i \cdot \color{blue}{\left(\frac{1}{\beta} \cdot \left(\frac{1}{\beta} \cdot \left(i + \alpha\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{1}{\beta} \cdot \left(\left(\alpha + i\right) \cdot \frac{1}{\beta}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+238}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.4e+238) 0.0625 (/ (* (+ beta alpha) 0.0) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+238) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.4d+238) then
        tmp = 0.0625d0
    else
        tmp = ((beta + alpha) * 0.0d0) / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+238) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.4e+238:
		tmp = 0.0625
	else:
		tmp = ((beta + alpha) * 0.0) / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.4e+238)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(beta + alpha) * 0.0) / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.4e+238)
		tmp = 0.0625;
	else
		tmp = ((beta + alpha) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.4e+238], 0.0625, N[(N[(N[(beta + alpha), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3999999999999998e238

   1. Initial program 19.5%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in i around inf 72.5%

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

   if 3.3999999999999998e238 < beta

   1. Initial program 0.0%

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

   3. Simplified 10.3%

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

   5. Taylor expanded in i around inf 52.2%

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

   6. Taylor expanded in i around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. distribute-lft-out 52.2%

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

      3. distribute-lft-in 52.2%

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

      4. metadata-eval 52.2%

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

      5. *-commutative 52.2%

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

   8. Simplified 52.2%

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

   9. Taylor expanded in i around 0 41.0%

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

   11. Simplified 41.0%

       \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right) \cdot 0}}{i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+238}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

3. Simplified 38.3%

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

5. Taylor expanded in i around inf 66.0%

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

6. Final simplification 66.0%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024051 
(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)))