Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 84.9%

Time: 14.6s

Alternatives: 8

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+192)
   (-
    (+ 0.0625 (* 0.0625 (/ (* 2.0 (+ beta alpha)) i)))
    (* 0.125 (/ (+ beta alpha) i)))
   (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+192) {
		tmp = (0.0625 + (0.0625 * ((2.0 * (beta + alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + 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 <= 6d+192) then
        tmp = (0.0625d0 + (0.0625d0 * ((2.0d0 * (beta + alpha)) / i))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = ((alpha + 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 <= 6e+192) {
		tmp = (0.0625 + (0.0625 * ((2.0 * (beta + alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+192:
		tmp = (0.0625 + (0.0625 * ((2.0 * (beta + alpha)) / i))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+192)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(2.0 * Float64(beta + alpha)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(alpha + 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 <= 6e+192)
		tmp = (0.0625 + (0.0625 * ((2.0 * (beta + alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = ((alpha + 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, 6e+192], N[(N[(0.0625 + N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e192

   1. Initial program 18.9%

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

   3. Simplified 41.7%

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

   5. Taylor expanded in i around inf 82.9%

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

      1. distribute-lft-out 82.9%

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

   7. Simplified 82.9%

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

   if 6e192 < beta

   1. Initial program 0.0%

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

   3. Simplified 4.9%

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

   5. Taylor expanded in beta around inf 25.7%

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

      1. unpow2 25.7%

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

   7. Applied egg-rr 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-/r* 45.5%

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

      3. associate-*l/ 78.0%

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

   9. Applied egg-rr 78.0%

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

       1. associate-/l* 78.2%

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

   11. Applied egg-rr 78.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.18 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.18e+198) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.18e+198) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.18d+198) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.18e+198) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.18e+198:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.18e+198)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.18e+198)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.18e+198], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.17999999999999998e198

   1. Initial program 18.7%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in i around inf 77.5%

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

   if 1.17999999999999998e198 < beta

   1. Initial program 0.0%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in beta around inf 26.8%

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

      1. unpow2 26.8%

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

   7. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

      2. associate-/r* 45.1%

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

      3. associate-*l/ 79.2%

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

   9. Applied egg-rr 79.2%

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

       1. associate-/l* 79.5%

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

   11. Applied egg-rr 79.5%

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

Alternative 3: 82.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.35e+196) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.35e+196) {
		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 <= 1.35d+196) 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 <= 1.35e+196) {
		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 <= 1.35e+196:
		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 <= 1.35e+196)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.35e+196)
		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, 1.35e+196], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.34999999999999998e196

   1. Initial program 18.7%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in i around inf 77.5%

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

   if 1.34999999999999998e196 < beta

   1. Initial program 0.0%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in beta around inf 26.8%

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

      1. unpow2 26.8%

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

   7. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

      2. associate-/r* 45.1%

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

      3. associate-*l/ 79.2%

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

   9. Applied egg-rr 79.2%

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

   10. Taylor expanded in alpha around 0 63.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4e+204) 0.0625 (/ (* i (/ alpha beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (i * (alpha / 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 <= 4d+204) then
        tmp = 0.0625d0
    else
        tmp = (i * (alpha / 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 <= 4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (i * (alpha / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4e+204:
		tmp = 0.0625
	else:
		tmp = (i * (alpha / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4e+204)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(alpha / 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 <= 4e+204)
		tmp = 0.0625;
	else
		tmp = (i * (alpha / 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, 4e+204], 0.0625, N[(N[(i * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999996e204

   1. Initial program 18.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in i around inf 77.2%

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

   if 3.99999999999999996e204 < beta

   1. Initial program 0.0%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in beta around inf 27.3%

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

      1. unpow2 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. *-commutative 27.3%

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

      2. associate-/r* 43.8%

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

      3. associate-*l/ 78.7%

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

   9. Applied egg-rr 78.7%

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

   10. Taylor expanded in alpha around inf 43.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4e+204) 0.0625 (/ (* alpha (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * (i / beta)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4d+204) then
        tmp = 0.0625d0
    else
        tmp = (alpha * (i / beta)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4e+204:
		tmp = 0.0625
	else:
		tmp = (alpha * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4e+204)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha * 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 <= 4e+204)
		tmp = 0.0625;
	else
		tmp = (alpha * (i / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4e+204], 0.0625, N[(N[(alpha * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999996e204

   1. Initial program 18.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in i around inf 77.2%

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

   if 3.99999999999999996e204 < beta

   1. Initial program 0.0%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in beta around inf 27.3%

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

      1. unpow2 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. *-commutative 27.3%

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

      2. associate-/r* 43.8%

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

      3. associate-*l/ 78.7%

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

   9. Applied egg-rr 78.7%

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

   10. Taylor expanded in alpha around inf 32.4%

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

       1. associate-/l* 43.9%

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

   12. Simplified 43.9%

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

Alternative 6: 74.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+228}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.2e+228) 0.0625 (* i (/ i (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+228) {
		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 <= 5.2d+228) 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 <= 5.2e+228) {
		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 <= 5.2e+228:
		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 <= 5.2e+228)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.2e+228)
		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, 5.2e+228], 0.0625, N[(i * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000015e228

   1. Initial program 17.9%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in i around inf 74.3%

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

   if 5.20000000000000015e228 < beta

   1. Initial program 0.0%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in beta around inf 33.6%

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

      1. unpow2 33.6%

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

   7. Applied egg-rr 33.6%

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

   8. Taylor expanded in alpha around 0 33.6%

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

Alternative 7: 74.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{+228}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.1e+228) 0.0625 (* i (/ alpha (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.1e+228) {
		tmp = 0.0625;
	} else {
		tmp = i * (alpha / (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 <= 4.1d+228) then
        tmp = 0.0625d0
    else
        tmp = i * (alpha / (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 <= 4.1e+228) {
		tmp = 0.0625;
	} else {
		tmp = i * (alpha / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.1e+228:
		tmp = 0.0625
	else:
		tmp = i * (alpha / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.1e+228)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(alpha / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.1e+228)
		tmp = 0.0625;
	else
		tmp = i * (alpha / (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, 4.1e+228], 0.0625, N[(i * N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1e228

   1. Initial program 17.9%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in i around inf 74.3%

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

   if 4.1e228 < beta

   1. Initial program 0.0%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in beta around inf 33.6%

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

      1. unpow2 33.6%

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

   7. Applied egg-rr 33.6%

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

   8. Taylor expanded in alpha around inf 33.6%

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

Alternative 8: 70.6% 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 15.6%

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

3. Simplified 35.4%

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

5. Taylor expanded in i around inf 66.0%

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

Reproduce

herbie shell --seed 2024135 
(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)))