Octave 3.8, jcobi/4

Percentage Accurate: 14.9% → 85.5%

Time: 14.4s

Alternatives: 7

Speedup: 115.0×

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: 14.9% 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: 85.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \left(i + \alpha\right)\right) \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.6e+157)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (* (/ 1.0 beta) (+ i alpha)) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+157) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((1.0 / beta) * (i + alpha)) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.6d+157) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((1.0d0 / beta) * (i + alpha)) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+157) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((1.0 / beta) * (i + alpha)) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.6e+157:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((1.0 / beta) * (i + alpha)) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.6e+157)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) * Float64(i + alpha)) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.6e+157)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((1.0 / beta) * (i + alpha)) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.6e+157], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000011e157

   1. Initial program 22.5%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 19.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 38.0

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

   8. Simplified 38.0%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. *-lowering-*.f64 80.3

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

   11. Simplified 80.3%

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

   if 2.60000000000000011e157 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 24.5

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

   5. Simplified 24.5%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 73.7

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

   7. Applied egg-rr 73.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 73.8

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

   9. Applied egg-rr 73.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \left(i + \alpha\right)\right) \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+156}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.4e+156)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ i beta) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+156) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.4d+156) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+156) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.4e+156:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.4e+156)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.4e+156)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.4e+156], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4000000000000001e156

   1. Initial program 22.5%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 19.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 38.0

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

   8. Simplified 38.0%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. *-lowering-*.f64 80.3

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

   11. Simplified 80.3%

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

   if 2.4000000000000001e156 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 24.5

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

   5. Simplified 24.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 N/A

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

      7. /-lowering-/.f64 73.8

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

   7. Applied egg-rr 73.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+156}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+155}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.6e+155)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+155) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i * (i / beta)) / 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 <= 2.6d+155) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i * (i / beta)) / 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 <= 2.6e+155) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.6e+155:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.6e+155)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / 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 <= 2.6e+155)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i * (i / beta)) / 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, 2.6e+155], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6000000000000002e155

   1. Initial program 22.5%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 19.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 38.0

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

   8. Simplified 38.0%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. *-lowering-*.f64 80.3

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

   11. Simplified 80.3%

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

   if 2.6000000000000002e155 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 24.5

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

   5. Simplified 24.5%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 73.7

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

   7. Applied egg-rr 73.7%

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

   8. Taylor expanded in i around inf

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

   10. Simplified 67.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+155}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+227}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.1e+227)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.1e+227) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.1d+227) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.1e+227) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.1e+227:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.1e+227)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.1e+227)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.1e+227], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.10000000000000015e227

   1. Initial program 20.2%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 17.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 34.2

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

   8. Simplified 34.2%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. *-lowering-*.f64 76.5

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

   11. Simplified 76.5%

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

   if 6.10000000000000015e227 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in i around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.2

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

   8. Simplified 49.2%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 53.5

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

   10. Applied egg-rr 53.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+227}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.2e+229) (+ 0.0625 (/ 0.015625 (* i i))) (/ 0.0 i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.2e+229) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 8.2d+229) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.2e+229) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.2e+229:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.2e+229)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.2e+229)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.2e+229], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2000000000000003e229

   1. Initial program 20.2%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 17.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{4 \cdot {i}^{2} - 1} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 34.2

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

   8. Simplified 34.2%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. *-lowering-*.f64 76.5

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

   11. Simplified 76.5%

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

   if 8.2000000000000003e229 < 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. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{256} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{\beta}{i} \cdot \frac{1}{8}} + \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\beta}{i}}, \frac{1}{8}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{\color{blue}{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{\color{blue}{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{16 \cdot \color{blue}{\left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      13. +-lowering-+.f64 49.5

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, 0.125, 0.0625\right) + -0.00390625 \cdot \frac{16 \cdot \left(\beta + \color{blue}{\left(\alpha + \beta\right)}\right)}{i} \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{i}, 0.125, 0.0625\right) + -0.00390625 \cdot \frac{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}{i}} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\beta}{i}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\beta}{i}} \]

       2. /-lowering-/.f64 49.5

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

   11. Simplified 49.5%

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

   12. Taylor expanded in i around 0

       \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \beta + \frac{1}{8} \cdot \beta}{i}} \]
   13. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{\beta \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}}{i} \]

       2. metadata-eval N/A

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

       3. mul0-rgt N/A

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

       4. /-lowering-/.f64 49.2

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

   14. Simplified 49.2%

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

Alternative 6: 74.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.3e+229) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.3e+229) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.3d+229) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.3e+229) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.3e+229:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.3e+229)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.3e+229)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.3e+229], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999999e229

   1. Initial program 20.2%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Simplified 76.0%

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

   if 2.2999999999999999e229 < 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. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{256} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{\beta}{i} \cdot \frac{1}{8}} + \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\beta}{i}}, \frac{1}{8}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{16 \cdot \beta + 16 \cdot \left(\alpha + \beta\right)}{i}} \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{\color{blue}{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{\color{blue}{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \frac{-1}{256} \cdot \frac{16 \cdot \color{blue}{\left(\beta + \left(\alpha + \beta\right)\right)}}{i} \]

      13. +-lowering-+.f64 49.5

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, 0.125, 0.0625\right) + -0.00390625 \cdot \frac{16 \cdot \left(\beta + \color{blue}{\left(\alpha + \beta\right)}\right)}{i} \]

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{i}, 0.125, 0.0625\right) + -0.00390625 \cdot \frac{16 \cdot \left(\beta + \left(\alpha + \beta\right)\right)}{i}} \]

   9. Taylor expanded in beta around inf

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\beta}{i}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{1}{8}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\beta}{i}} \]

       2. /-lowering-/.f64 49.5

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

   11. Simplified 49.5%

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

   12. Taylor expanded in i around 0

       \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \beta + \frac{1}{8} \cdot \beta}{i}} \]
   13. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{\beta \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}}{i} \]

       2. metadata-eval N/A

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

       3. mul0-rgt N/A

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

       4. /-lowering-/.f64 49.2

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

   14. Simplified 49.2%

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

Alternative 7: 70.6% accurate, 115.0× 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 18.5%

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

3. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
4. Step-by-step derivation

5. Simplified 69.9%

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

Reproduce

herbie shell --seed 2024197 
(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)))