Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 86.0%

Time: 13.5s

Alternatives: 10

Speedup: 53.0×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.3% 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: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.15 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\alpha + \left(\beta + \left(1 + i \cdot 2\right)\right)}}{\frac{\alpha + \left(\beta + \left(i \cdot 2 + -1\right)\right)}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.15e+125)
   0.0625
   (/
    (/ (+ i alpha) (+ alpha (+ beta (+ 1.0 (* i 2.0)))))
    (/ (+ alpha (+ beta (+ (* i 2.0) -1.0))) i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.15e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / ((alpha + (beta + ((i * 2.0) + -1.0))) / i);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.15d+125) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / (alpha + (beta + (1.0d0 + (i * 2.0d0))))) / ((alpha + (beta + ((i * 2.0d0) + (-1.0d0)))) / i)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.15e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / ((alpha + (beta + ((i * 2.0) + -1.0))) / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.15e+125:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / ((alpha + (beta + ((i * 2.0) + -1.0))) / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.15e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(beta + Float64(1.0 + Float64(i * 2.0))))) / Float64(Float64(alpha + Float64(beta + Float64(Float64(i * 2.0) + -1.0))) / i));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.15e+125)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / ((alpha + (beta + ((i * 2.0) + -1.0))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.15e+125], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(1.0 + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1500000000000001e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 3.1500000000000001e125 < beta

   1. Initial program 2.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 72.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right)} + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{\left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \left(\beta + \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{\left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - \color{blue}{1}\right)}{i}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(1 + i \cdot 2\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(1 + 2 \cdot i\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \left(2 \cdot i\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \left(i \cdot 2\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)\right), \color{blue}{i}\right)\right) \]

   9. Applied egg-rr 72.9%

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

Alternative 2: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) + \left(1 + i \cdot 2\right)} \cdot \frac{i}{\left(\beta + \alpha\right) + \left(i \cdot 2 + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+125)
   0.0625
   (*
    (/ (+ i alpha) (+ (+ beta alpha) (+ 1.0 (* i 2.0))))
    (/ i (+ (+ beta alpha) (+ (* i 2.0) -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / ((beta + alpha) + ((i * 2.0) + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.3d+125) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / ((beta + alpha) + (1.0d0 + (i * 2.0d0)))) * (i / ((beta + alpha) + ((i * 2.0d0) + (-1.0d0))))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / ((beta + alpha) + ((i * 2.0) + -1.0)));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+125:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / ((beta + alpha) + ((i * 2.0) + -1.0)))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(Float64(beta + alpha) + Float64(1.0 + Float64(i * 2.0)))) * Float64(i / Float64(Float64(beta + alpha) + Float64(Float64(i * 2.0) + -1.0))));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / ((beta + alpha) + ((i * 2.0) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+125], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(1.0 + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3000000000000003e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 5.3000000000000003e125 < beta

   1. Initial program 2.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 72.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) + \left(1 + i \cdot 2\right)} \cdot \frac{i}{\left(\beta + \alpha\right) + \left(i \cdot 2 + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\alpha + \left(\beta + \left(1 + i \cdot 2\right)\right)}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3e+125)
   0.0625
   (/ (/ (+ i alpha) (+ alpha (+ beta (+ 1.0 (* i 2.0))))) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / (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 <= 3d+125) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / (alpha + (beta + (1.0d0 + (i * 2.0d0))))) / (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 <= 3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3e+125:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(beta + Float64(1.0 + Float64(i * 2.0))))) / Float64(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 <= 3e+125)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / (alpha + (beta + (1.0 + (i * 2.0))))) / (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, 3e+125], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(1.0 + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.00000000000000015e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 3.00000000000000015e125 < beta

   1. Initial program 2.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 72.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)}\right), \color{blue}{\left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)\right)\right), \left(\frac{\color{blue}{\left(\alpha + \beta\right)} + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \left(\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{\left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \left(\beta + \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \color{blue}{\left(i \cdot 2 - 1\right)}}{i}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(i \cdot 2 + 1\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - \color{blue}{1}\right)}{i}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(1 + i \cdot 2\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(1 + 2 \cdot i\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \left(2 \cdot i\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \left(i \cdot 2\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \left(\frac{\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) + \left(i \cdot 2 - 1\right)\right), \color{blue}{i}\right)\right) \]

   9. Applied egg-rr 72.9%

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

   10. Taylor expanded in beta around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \color{blue}{\left(\frac{\beta}{i}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 71.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, 2\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\beta, \color{blue}{i}\right)\right) \]

   12. Simplified 71.0%

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

Alternative 4: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) + \left(1 + i \cdot 2\right)} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+125)
   0.0625
   (* (/ (+ i alpha) (+ (+ beta alpha) (+ 1.0 (* i 2.0)))) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.3d+125) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / ((beta + alpha) + (1.0d0 + (i * 2.0d0)))) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+125:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(Float64(beta + alpha) + Float64(1.0 + Float64(i * 2.0)))) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / ((beta + alpha) + (1.0 + (i * 2.0)))) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+125], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(1.0 + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3000000000000003e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 5.3000000000000003e125 < beta

   1. Initial program 2.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(\alpha + i\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

   5. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. difference-of-sqr-1 N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 72.9%

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

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), 1\right)\right)\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 2\right), 1\right)\right)\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   10. Simplified 71.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) + \left(1 + i \cdot 2\right)} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+250}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+125)
   0.0625
   (if (<= beta 7.5e+250)
     (* i (/ (/ i beta) beta))
     (* (/ i beta) (/ alpha beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else if (beta <= 7.5e+250) {
		tmp = i * ((i / beta) / beta);
	} 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 <= 5.3d+125) then
        tmp = 0.0625d0
    else if (beta <= 7.5d+250) then
        tmp = i * ((i / beta) / beta)
    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 <= 5.3e+125) {
		tmp = 0.0625;
	} else if (beta <= 7.5e+250) {
		tmp = i * ((i / beta) / beta);
	} 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 <= 5.3e+125:
		tmp = 0.0625
	elif beta <= 7.5e+250:
		tmp = i * ((i / beta) / beta)
	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 <= 5.3e+125)
		tmp = 0.0625;
	elseif (beta <= 7.5e+250)
		tmp = Float64(i * Float64(Float64(i / beta) / beta));
	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 <= 5.3e+125)
		tmp = 0.0625;
	elseif (beta <= 7.5e+250)
		tmp = i * ((i / beta) / beta);
	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, 5.3e+125], 0.0625, If[LessEqual[beta, 7.5e+250], N[(i * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.3000000000000003e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 5.3000000000000003e125 < beta < 7.4999999999999997e250

   1. Initial program 2.8%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 22.1%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 23.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 23.4%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i \cdot \left(\alpha + i\right)}{\beta}\right), \color{blue}{\beta}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \beta\right), \beta\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \beta\right), \beta\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \beta\right), \beta\right) \]

      6. +-lowering-+.f64 49.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 49.1%

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

   10. Taylor expanded in i around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \color{blue}{i}\right), \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 47.4%

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

       1. associate-/l* N/A

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

       2. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{\frac{i}{\beta}}{\beta}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\left(\frac{i}{\beta}\right), \color{blue}{\beta}\right)\right) \]

       5. /-lowering-/.f64 54.0%

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \beta\right)\right) \]

   14. Applied egg-rr 54.0%

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

   if 7.4999999999999997e250 < beta

   1. Initial program 0.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 57.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 57.8%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      7. /-lowering-/.f64 88.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 88.3%

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

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\alpha}, \beta\right), \mathsf{/.f64}\left(i, \beta\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 76.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+250}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.35 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.35e+125) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.35e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / 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 <= 3.35d+125) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / 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 <= 3.35e+125) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.35e+125:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.35e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(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 <= 3.35e+125)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / 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, 3.35e+125], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3500000000000002e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 3.3500000000000002e125 < beta

   1. Initial program 2.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      7. /-lowering-/.f64 70.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 70.6%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\beta}\right), \color{blue}{\left(\frac{\beta}{i}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{\color{blue}{\beta}}{i}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{\beta}{i}\right)\right) \]

       6. /-lowering-/.f64 70.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(\beta, \color{blue}{i}\right)\right) \]

   11. Applied egg-rr 70.6%

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

Alternative 7: 84.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.2e+125) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.2e+125) {
		tmp = 0.0625;
	} 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 <= 4.2d+125) then
        tmp = 0.0625d0
    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 <= 4.2e+125) {
		tmp = 0.0625;
	} 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 <= 4.2e+125:
		tmp = 0.0625
	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 <= 4.2e+125)
		tmp = 0.0625;
	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 <= 4.2e+125)
		tmp = 0.0625;
	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, 4.2e+125], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.2000000000000001e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 4.2000000000000001e125 < beta

   1. Initial program 2.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      7. /-lowering-/.f64 70.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 70.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+125) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.3d+125) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+125:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+125], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3000000000000003e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 5.3000000000000003e125 < beta

   1. Initial program 2.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 33.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      7. /-lowering-/.f64 70.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 70.6%

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

   10. Taylor expanded in i around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{i}, \beta\right), \mathsf{/.f64}\left(i, \beta\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 64.3%

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

Alternative 9: 76.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+125}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+125) 0.0625 (* i (/ (/ i beta) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+125) {
		tmp = 0.0625;
	} 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 <= 5.3d+125) then
        tmp = 0.0625d0
    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 <= 5.3e+125) {
		tmp = 0.0625;
	} 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 <= 5.3e+125:
		tmp = 0.0625
	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 <= 5.3e+125)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(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 <= 5.3e+125)
		tmp = 0.0625;
	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, 5.3e+125], 0.0625, N[(i * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3000000000000003e125

   1. Initial program 18.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 37.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 82.1%

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

   if 5.3000000000000003e125 < beta

   1. Initial program 2.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 33.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 33.0%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i \cdot \left(\alpha + i\right)}{\beta}\right), \color{blue}{\beta}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \beta\right), \beta\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \beta\right), \beta\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(i + \alpha\right)\right), \beta\right), \beta\right) \]

      6. +-lowering-+.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 54.8%

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

   10. Taylor expanded in i around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \color{blue}{i}\right), \beta\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 50.6%

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

       1. associate-/l* N/A

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

       2. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{\frac{i}{\beta}}{\beta}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\left(\frac{i}{\beta}\right), \color{blue}{\beta}\right)\right) \]

       5. /-lowering-/.f64 55.6%

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \beta\right)\right) \]

   14. Applied egg-rr 55.6%

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

Alternative 10: 71.0% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.9%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

3. Simplified 32.8%

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

5. Taylor expanded in i around inf

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

7. Simplified 68.2%

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

Reproduce

herbie shell --seed 2024154 
(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)))