Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 82.8%

Time: 14.5s

Alternatives: 7

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: 16.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.95e+173) 0.0625 (pow (/ i beta) 2.0)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.95e+173) {
		tmp = 0.0625;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.95d+173) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.95e+173) {
		tmp = 0.0625;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.95e+173:
		tmp = 0.0625
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.95e+173)
		tmp = 0.0625;
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.95e+173)
		tmp = 0.0625;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.95e+173], 0.0625, N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9499999999999999e173

   1. Initial program 19.8%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in i around inf 78.1%

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

   if 1.9499999999999999e173 < 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 14.6%

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

   5. Taylor expanded in alpha around 0 14.6%

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

   6. Taylor expanded in alpha around 0 14.6%

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

   7. Taylor expanded in beta around inf 26.7%

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

      1. unpow2 26.7%

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

      2. unpow2 26.7%

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

      3. times-frac 66.2%

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

      4. unpow1 66.2%

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

      5. pow-plus 66.2%

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

      6. metadata-eval 66.2%

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

   9. Simplified 66.2%

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

Alternative 2: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_2 + \beta \cdot \alpha\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (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 0.1)
     t_3
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* 2.0 alpha) (* beta 2.0)) i)))
      (* 0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
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 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: 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 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * (((2.0d0 * alpha) + (beta * 2.0d0)) / i))) - (0.125d0 * ((beta + alpha) / i))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
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 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
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 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
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 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(2.0 * alpha) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
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 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 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, 0.1], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * alpha), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * 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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.7%

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

   if 0.10000000000000001 < (/.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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.8%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in i around inf 73.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \end{array} \]

FPCore

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

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
}

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 + (0.0625d0 * (((2.0d0 * alpha) + (beta * 2.0d0)) / i))) - (0.125d0 * ((beta + alpha) / i))
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i))

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(2.0 * alpha) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)))
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (((2.0 * alpha) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * alpha), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

3. Simplified 40.6%

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

5. Taylor expanded in i around inf 75.0%

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

6. Final simplification 75.0%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
7. Add Preprocessing

Alternative 4: 74.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{i}{\beta} + 1\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.2e+233) 0.0625 (* 0.25 (+ (+ (/ i beta) 1.0) -1.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+233) {
		tmp = 0.0625;
	} else {
		tmp = 0.25 * (((i / beta) + 1.0) + -1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.2d+233) then
        tmp = 0.0625d0
    else
        tmp = 0.25d0 * (((i / beta) + 1.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+233) {
		tmp = 0.0625;
	} else {
		tmp = 0.25 * (((i / beta) + 1.0) + -1.0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.2e+233:
		tmp = 0.0625
	else:
		tmp = 0.25 * (((i / beta) + 1.0) + -1.0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.2e+233)
		tmp = 0.0625;
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(i / beta) + 1.0) + -1.0));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.2e+233)
		tmp = 0.0625;
	else
		tmp = 0.25 * (((i / beta) + 1.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.2e+233], 0.0625, N[(0.25 * N[(N[(N[(i / beta), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000013e233

   1. Initial program 18.4%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in i around inf 75.2%

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

   if 5.20000000000000013e233 < 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 15.8%

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

   5. Taylor expanded in i around inf 39.6%

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

   6. Taylor expanded in beta around inf 8.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{i}{\beta}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{i}{\beta}} \cdot \sqrt[3]{\frac{i}{\beta}}\right) \cdot \sqrt[3]{\frac{i}{\beta}}\right)} \]

      2. pow3 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\sqrt[3]{\frac{i}{\beta}}\right)}^{3}} \]

   8. Applied egg-rr 8.2%

      \[\leadsto 0.25 \cdot \color{blue}{{\left(\sqrt[3]{\frac{i}{\beta}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{i}{\beta}} \]

      2. expm1-log1p-u 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{\beta}\right)\right)} \]

      3. log1p-define 42.0%

         \[\leadsto 0.25 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{\beta}\right)}\right) \]

      4. expm1-undefine 42.0%

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

      5. add-exp-log 42.0%

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

   10. Applied egg-rr 42.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(1 + \frac{i}{\beta}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{i}{\beta} + 1\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.72 \cdot 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + \left(\frac{i}{\beta} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.72e+233) 0.0625 (* 0.25 (+ 1.0 (+ (/ i beta) -1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.72e+233) {
		tmp = 0.0625;
	} else {
		tmp = 0.25 * (1.0 + ((i / beta) + -1.0));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.72d+233) then
        tmp = 0.0625d0
    else
        tmp = 0.25d0 * (1.0d0 + ((i / beta) + (-1.0d0)))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.72e+233) {
		tmp = 0.0625;
	} else {
		tmp = 0.25 * (1.0 + ((i / beta) + -1.0));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.72e+233:
		tmp = 0.0625
	else:
		tmp = 0.25 * (1.0 + ((i / beta) + -1.0))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.72e+233)
		tmp = 0.0625;
	else
		tmp = Float64(0.25 * Float64(1.0 + Float64(Float64(i / beta) + -1.0)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.72e+233)
		tmp = 0.0625;
	else
		tmp = 0.25 * (1.0 + ((i / beta) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.72e+233], 0.0625, N[(0.25 * N[(1.0 + N[(N[(i / beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.7200000000000001e233

   1. Initial program 18.4%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in i around inf 75.2%

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

   if 1.7200000000000001e233 < 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 15.8%

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

   5. Taylor expanded in i around inf 39.6%

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

   6. Taylor expanded in beta around inf 8.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{i}{\beta}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{i}{\beta}} \cdot \sqrt[3]{\frac{i}{\beta}}\right) \cdot \sqrt[3]{\frac{i}{\beta}}\right)} \]

      2. pow3 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\sqrt[3]{\frac{i}{\beta}}\right)}^{3}} \]

   8. Applied egg-rr 8.2%

      \[\leadsto 0.25 \cdot \color{blue}{{\left(\sqrt[3]{\frac{i}{\beta}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{i}{\beta}} \]

      2. expm1-log1p-u 8.2%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{\beta}\right)\right)} \]

      3. log1p-define 42.0%

         \[\leadsto 0.25 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{\beta}\right)}\right) \]

      4. expm1-undefine 42.0%

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

      5. add-exp-log 42.0%

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

   10. Applied egg-rr 42.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(1 + \frac{i}{\beta}\right) - 1\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 42.0%

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

   12. Simplified 42.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(\frac{i}{\beta} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.72 \cdot 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + \left(\frac{i}{\beta} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+258}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.8e+258) 0.0625 (/ (* (+ beta alpha) 0.0) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+258) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.8d+258) then
        tmp = 0.0625d0
    else
        tmp = ((beta + alpha) * 0.0d0) / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+258) {
		tmp = 0.0625;
	} else {
		tmp = ((beta + alpha) * 0.0) / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.8e+258:
		tmp = 0.0625
	else:
		tmp = ((beta + alpha) * 0.0) / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.8e+258)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(beta + alpha) * 0.0) / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.8e+258)
		tmp = 0.0625;
	else
		tmp = ((beta + alpha) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.8e+258], 0.0625, N[(N[(N[(beta + alpha), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.79999999999999982e258

   1. Initial program 17.8%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in i around inf 73.6%

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

   if 2.79999999999999982e258 < 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 16.7%

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

   5. Taylor expanded in i around inf 63.9%

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

      1. add-log-exp 63.4%

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

      2. cancel-sign-sub-inv 63.4%

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

      3. +-commutative 63.4%

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

      4. fma-define 63.4%

         \[\leadsto \log \left(e^{\color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, 0.0625\right)} + \left(-0.125\right) \cdot \frac{\alpha + \beta}{i}}\right) \]

      5. distribute-lft-out 63.4%

         \[\leadsto \log \left(e^{\mathsf{fma}\left(0.0625, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, 0.0625\right) + \left(-0.125\right) \cdot \frac{\alpha + \beta}{i}}\right) \]

      6. metadata-eval 63.4%

         \[\leadsto \log \left(e^{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + \color{blue}{-0.125} \cdot \frac{\alpha + \beta}{i}}\right) \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}}\right)} \]

   8. Taylor expanded in i around 0 51.9%

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

      1. distribute-rgt-out 51.9%

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

      2. +-commutative 51.9%

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

      3. metadata-eval 51.9%

         \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{0}}{i} \]

   10. Simplified 51.9%

       \[\leadsto \color{blue}{\frac{\left(\beta + \alpha\right) \cdot 0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.8% 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 17.0%

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

3. Simplified 40.6%

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

5. Taylor expanded in i around inf 70.9%

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

Reproduce

herbie shell --seed 2024139 
(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)))