Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 84.1%

Time: 17.5s

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: 16.3% 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: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\frac{\beta}{\sqrt{i + \alpha}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.4e+122)
   0.0625
   (pow (/ (sqrt i) (/ beta (sqrt (+ i alpha)))) 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.4e+122) {
		tmp = 0.0625;
	} else {
		tmp = pow((sqrt(i) / (beta / sqrt((i + alpha)))), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.4d+122) then
        tmp = 0.0625d0
    else
        tmp = (sqrt(i) / (beta / sqrt((i + alpha)))) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.4e+122) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow((Math.sqrt(i) / (beta / Math.sqrt((i + alpha)))), 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.4e+122:
		tmp = 0.0625
	else:
		tmp = math.pow((math.sqrt(i) / (beta / math.sqrt((i + alpha)))), 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.4e+122)
		tmp = 0.0625;
	else
		tmp = Float64(sqrt(i) / Float64(beta / sqrt(Float64(i + alpha)))) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.4e+122)
		tmp = 0.0625;
	else
		tmp = (sqrt(i) / (beta / sqrt((i + alpha)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.4e+122], 0.0625, N[Power[N[(N[Sqrt[i], $MachinePrecision] / N[(beta / N[Sqrt[N[(i + alpha), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.40000000000000024e122

   1. Initial program 23.5%

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

      1. associate-/l/ 19.6%

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

      2. associate-*l* 19.5%

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

      3. times-frac 29.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in i around inf 78.1%

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

   if 6.40000000000000024e122 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 31.7%

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

      2. +-commutative 31.7%

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

   6. Simplified 31.7%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

      3. sqrt-div 31.6%

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

      4. sqrt-div 31.6%

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

      5. unpow2 31.6%

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

      6. sqrt-prod 68.8%

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

      7. add-sqr-sqrt 69.1%

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

   8. Applied egg-rr 69.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\frac{\beta}{\sqrt{i + \alpha}}}\right)}^{2}\\ \end{array} \]

Alternative 2: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\beta} \cdot \sqrt{i + \alpha}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+122)
   0.0625
   (pow (* (/ (sqrt i) beta) (sqrt (+ i alpha))) 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = 0.0625;
	} else {
		tmp = pow(((sqrt(i) / beta) * sqrt((i + alpha))), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 7.5d+122) then
        tmp = 0.0625d0
    else
        tmp = ((sqrt(i) / beta) * sqrt((i + alpha))) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+122) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow(((Math.sqrt(i) / beta) * Math.sqrt((i + alpha))), 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+122:
		tmp = 0.0625
	else:
		tmp = math.pow(((math.sqrt(i) / beta) * math.sqrt((i + alpha))), 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+122)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(sqrt(i) / beta) * sqrt(Float64(i + alpha))) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+122)
		tmp = 0.0625;
	else
		tmp = ((sqrt(i) / beta) * sqrt((i + alpha))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.5e+122], 0.0625, N[Power[N[(N[(N[Sqrt[i], $MachinePrecision] / beta), $MachinePrecision] * N[Sqrt[N[(i + alpha), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5000000000000002e122

   1. Initial program 23.5%

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

      1. associate-/l/ 19.6%

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

      2. associate-*l* 19.5%

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

      3. times-frac 29.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in i around inf 78.1%

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

   if 7.5000000000000002e122 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 31.7%

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

      2. +-commutative 31.7%

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

   6. Simplified 31.7%

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

      1. add-sqr-sqrt 31.6%

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

      2. sqrt-div 31.6%

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

      3. sqrt-div 31.7%

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

      4. unpow2 31.7%

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

      5. sqrt-prod 31.7%

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

      6. add-sqr-sqrt 31.7%

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

      7. sqrt-div 31.7%

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

      8. sqrt-div 31.6%

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

      9. unpow2 31.6%

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

      10. sqrt-prod 69.1%

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

      11. add-sqr-sqrt 69.1%

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

   8. Applied egg-rr 69.1%

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

      1. unpow2 69.1%

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

      2. associate-/r/ 69.1%

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

      3. +-commutative 69.1%

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

   10. Simplified 69.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{i}}{\beta} \cdot \sqrt{i + \alpha}\right)}^{2}\\ \end{array} \]

Alternative 3: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \beta \cdot \alpha\right)}{t_1}}{t_1 + -1}\\ \mathbf{if}\;t_3 \leq 0.06249999999995:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.06249999999995)
     t_3
     (-
      (+
       0.0625
       (*
        0.25
        (/
         (- (* 0.25 (+ (* alpha 2.0) (* beta 2.0))) (* (+ beta alpha) 0.25))
         i)))
      (* 0.0625 (/ (+ beta alpha) i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.06249999999995) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (beta + alpha) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (beta + alpha))
    t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.06249999999995d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.25d0 * (((0.25d0 * ((alpha * 2.0d0) + (beta * 2.0d0))) - ((beta + alpha) * 0.25d0)) / i))) - (0.0625d0 * ((beta + alpha) / i))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.06249999999995) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (beta + alpha))
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.06249999999995:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.06249999999995)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(Float64(0.25 * Float64(Float64(alpha * 2.0) + Float64(beta * 2.0))) - Float64(Float64(beta + alpha) * 0.25)) / i))) - Float64(0.0625 * Float64(Float64(beta + alpha) / i)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (beta + alpha));
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.06249999999995)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.06249999999995], t$95$3, N[(N[(0.0625 + N[(0.25 * N[(N[(N[(0.25 * N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.062499999999949998

   1. Initial program 98.8%

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

   if 0.062499999999949998 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 13.7%

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

   2. Taylor expanded in i around inf 31.5%

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Taylor expanded in i around inf 76.6%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1} \leq 0.06249999999995:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]

Alternative 4: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.5e+123) 0.0625 (pow (/ i beta) 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.5e+123) {
		tmp = 0.0625;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 1.5d+123) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.5e+123) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.5e+123:
		tmp = 0.0625
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.5e+123)
		tmp = 0.0625;
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.5e+123)
		tmp = 0.0625;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.5e+123], 0.0625, N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.50000000000000004e123

   1. Initial program 23.5%

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

      1. associate-/l/ 19.6%

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

      2. associate-*l* 19.5%

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

      3. times-frac 29.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in i around inf 78.1%

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

   if 1.50000000000000004e123 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in beta around inf 29.5%

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

      1. associate-/l* 31.7%

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

      2. +-commutative 31.7%

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

   6. Simplified 31.7%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

      3. sqrt-div 31.6%

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

      4. sqrt-div 31.6%

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

      5. unpow2 31.6%

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

      6. sqrt-prod 68.8%

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

      7. add-sqr-sqrt 69.1%

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

   8. Applied egg-rr 69.1%

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

   9. Taylor expanded in i around inf 63.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]

Alternative 5: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (-
  (+
   0.0625
   (*
    0.25
    (/ (- (* 0.25 (+ (* alpha 2.0) (* beta 2.0))) (* (+ beta alpha) 0.25)) i)))
  (* 0.0625 (/ (+ beta alpha) i))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
}

Fortran

NOTE: alpha and beta 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 + (0.25d0 * (((0.25d0 * ((alpha * 2.0d0) + (beta * 2.0d0))) - ((beta + alpha) * 0.25d0)) / i))) - (0.0625d0 * ((beta + alpha) / i))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	return (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(Float64(0.25 * Float64(Float64(alpha * 2.0) + Float64(beta * 2.0))) - Float64(Float64(beta + alpha) * 0.25)) / i))) - Float64(0.0625 * Float64(Float64(beta + alpha) / i)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.25 * (((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((beta + alpha) * 0.25)) / i))) - (0.0625 * ((beta + alpha) / i));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.25 * N[(N[(N[(0.25 * N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

2. Taylor expanded in i around inf 30.5%

   \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2} + i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

3. Taylor expanded in i around inf 73.0%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

4. Final simplification 73.0%

   \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\beta + \alpha\right) \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i} \]

Alternative 6: 77.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \left(0.0625 + t_0\right) - t_0 \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i)))) (- (+ 0.0625 t_0) t_0)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_0;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = 0.125d0 * (beta / i)
    code = (0.0625d0 + t_0) - t_0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	return (0.0625 + t_0) - t_0

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	return Float64(Float64(0.0625 + t_0) - t_0)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = (0.0625 + t_0) - t_0;
end

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. associate-/l/ 15.3%

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

   2. associate-*l* 15.2%

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

   3. times-frac 22.8%

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

3. Simplified 38.2%

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

4. Taylor expanded in i around inf 72.9%

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

5. Taylor expanded in alpha around 0 70.0%

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

6. Taylor expanded in i around 0 70.0%

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

7. Taylor expanded in alpha around 0 71.2%

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

8. Final simplification 71.2%

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

Alternative 7: 74.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+250}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\beta - \left(\beta + \alpha\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.3e+250) 0.0625 (/ (* 0.125 (- beta (+ beta alpha))) i)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.3e+250) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 1.3d+250) then
        tmp = 0.0625d0
    else
        tmp = (0.125d0 * (beta - (beta + alpha))) / i
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.3e+250) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.3e+250:
		tmp = 0.0625
	else:
		tmp = (0.125 * (beta - (beta + alpha))) / i
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.3e+250)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(0.125 * Float64(beta - Float64(beta + alpha))) / i);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.3e+250)
		tmp = 0.0625;
	else
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.3e+250], 0.0625, N[(N[(0.125 * N[(beta - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000006e250

   1. Initial program 20.0%

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

      1. associate-/l/ 16.7%

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

      2. associate-*l* 16.6%

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

      3. times-frac 24.9%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in i around inf 70.9%

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

   if 1.30000000000000006e250 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 9.5%

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

   4. Taylor expanded in i around inf 55.5%

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

   5. Taylor expanded in alpha around 0 55.1%

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

   6. Taylor expanded in i around 0 49.0%

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

      1. distribute-lft-out-- 49.0%

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

   8. Simplified 49.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+250}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\beta - \left(\beta + \alpha\right)\right)}{i}\\ \end{array} \]

Alternative 8: 70.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta 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;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. associate-/l/ 15.3%

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

   2. associate-*l* 15.2%

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

   3. times-frac 22.8%

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

3. Simplified 38.2%

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

4. Taylor expanded in i around inf 65.9%

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

5. Final simplification 65.9%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023308 
(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)))