Octave 3.8, jcobi/4

Percentage Accurate: 15.4% → 86.1%

Time: 11.4s

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: 15.4% 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: 86.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+170)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ i beta) (/ (+ alpha i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+170) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * ((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 <= 1.55d+170) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (i / beta) * ((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 <= 1.55e+170) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.55e+170:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+170)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + 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 <= 1.55e+170)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (i / beta) * ((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, 1.55e+170], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55e170

   1. Initial program 19.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 20.1%

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

   6. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 81.3%

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

   if 1.55e170 < 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 23.1

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

   5. Applied rewrites 23.1%

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

   7. Applied rewrites 78.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+170)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+170) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.55d+170) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+170) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.55e+170:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+170)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.55e+170)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.55e+170], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55e170

   1. Initial program 19.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 20.1%

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

   6. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 81.3%

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

   if 1.55e170 < 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 23.1

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

   5. Applied rewrites 23.1%

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.6%

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

Alternative 3: 76.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+214}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.65e+214)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ alpha beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.65e+214) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (alpha / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.65d+214) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (alpha / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.65e+214) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (alpha / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.65e+214:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (alpha / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.65e+214)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(alpha / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.65e+214)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (alpha / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.65e+214], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.65000000000000006e214

   1. Initial program 18.4%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 79.4%

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

   if 1.65000000000000006e214 < 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 40.8%

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

Alternative 4: 75.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta \cdot \beta} \cdot i\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.8e+244)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ (+ alpha i) (* beta beta)) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.8e+244) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((alpha + i) / (beta * beta)) * 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 <= 6.8d+244) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = ((alpha + i) / (beta * beta)) * 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 <= 6.8e+244) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((alpha + i) / (beta * beta)) * i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.8e+244:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = ((alpha + i) / (beta * beta)) * i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.8e+244)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(beta * beta)) * 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 <= 6.8e+244)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = ((alpha + i) / (beta * beta)) * 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, 6.8e+244], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8000000000000002e244

   1. Initial program 17.8%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 77.2%

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

   if 6.8000000000000002e244 < 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   7. Applied rewrites 37.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta \cdot \beta} \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.8e+244)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (/ (* alpha i) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.8e+244) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (alpha * 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 <= 6.8d+244) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (alpha * 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 <= 6.8e+244) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (alpha * i) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.8e+244:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (alpha * i) / (beta * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.8e+244)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(alpha * i) / Float64(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 <= 6.8e+244)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (alpha * 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, 6.8e+244], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(alpha * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8000000000000002e244

   1. Initial program 17.8%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 77.2%

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

   if 6.8000000000000002e244 < 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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 36.9%

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

Alternative 6: 71.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{0.015625}{i \cdot i} + 0.0625 \end{array} \]

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.015625 / (i * i)) + 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 = (0.015625d0 / (i * i)) + 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (0.015625 / (i * i)) + 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.015625 / (i * i)) + 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.015625 / Float64(i * i)) + 0.0625)
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.015625 / (i * i)) + 0.0625;
end

Wolfram

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

Derivation

1. Initial program 16.2%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. unpow2 N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-fma.f64 N/A

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

   20. unpow2 N/A

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

5. Applied rewrites 17.0%

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

6. Taylor expanded in i around inf

   \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 66.9%

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

9. Taylor expanded in beta around 0

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

11. Applied rewrites 70.7%

    \[\leadsto \frac{0.015625}{i \cdot i} + 0.0625 \]
12. Add Preprocessing

Alternative 7: 71.7% 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 16.2%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 70.6%

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

Reproduce

herbie shell --seed 2024235 
(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)))