Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 83.0%

Time: 18.0s

Alternatives: 5

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: 16.0% 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: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \frac{\beta}{\sqrt{i + \alpha}}\\ t_2 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\beta + i\right)}{\beta + i \cdot 2} \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}{\left(\beta + i\right) + \alpha}}}{t_2 \cdot t_2 + -1}\\ \mathbf{elif}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1} \cdot \frac{i}{t_1}\\ \end{array} \end{array} \]

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 (* 0.125 (/ beta i)))
        (t_1 (/ beta (sqrt (+ i alpha))))
        (t_2 (+ (* i 2.0) (+ beta alpha))))
   (if (<= beta 5.8e+71)
     0.0625
     (if (<= beta 2.25e+91)
       (/
        (*
         (/ (* i (+ beta i)) (+ beta (* i 2.0)))
         (/ i (/ (+ beta (fma i 2.0 alpha)) (+ (+ beta i) alpha))))
        (+ (* t_2 t_2) -1.0))
       (if (<= beta 1.56e+218)
         (- (+ 0.0625 t_0) t_0)
         (* (/ 1.0 t_1) (/ i t_1)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = beta / sqrt((i + alpha));
	double t_2 = (i * 2.0) + (beta + alpha);
	double tmp;
	if (beta <= 5.8e+71) {
		tmp = 0.0625;
	} else if (beta <= 2.25e+91) {
		tmp = (((i * (beta + i)) / (beta + (i * 2.0))) * (i / ((beta + fma(i, 2.0, alpha)) / ((beta + i) + alpha)))) / ((t_2 * t_2) + -1.0);
	} else if (beta <= 1.56e+218) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = (1.0 / t_1) * (i / t_1);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	t_1 = Float64(beta / sqrt(Float64(i + alpha)))
	t_2 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5.8e+71)
		tmp = 0.0625;
	elseif (beta <= 2.25e+91)
		tmp = Float64(Float64(Float64(Float64(i * Float64(beta + i)) / Float64(beta + Float64(i * 2.0))) * Float64(i / Float64(Float64(beta + fma(i, 2.0, alpha)) / Float64(Float64(beta + i) + alpha)))) / Float64(Float64(t_2 * t_2) + -1.0));
	elseif (beta <= 1.56e+218)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(1.0 / t_1) * Float64(i / t_1));
	end
	return 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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / N[Sqrt[N[(i + alpha), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.8e+71], 0.0625, If[LessEqual[beta, 2.25e+91], N[(N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + i), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.56e+218], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 5.80000000000000014e71

   1. Initial program 20.3%

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

      1. associate-/l/ 18.6%

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

      2. associate-*l* 18.6%

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

      3. times-frac 28.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in i around inf 82.1%

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

   if 5.80000000000000014e71 < beta < 2.25e91

   1. Initial program 34.2%

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

      1. times-frac 66.3%

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

      2. associate-+l+ 66.3%

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

      3. +-commutative 66.3%

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

      4. *-commutative 66.3%

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

      5. associate-+r+ 66.3%

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

      6. +-commutative 66.3%

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

      7. fma-def 66.3%

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

      8. +-commutative 66.3%

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

      9. +-commutative 66.3%

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

      10. *-commutative 66.3%

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

      11. fma-udef 66.3%

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

      12. +-commutative 66.3%

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

      13. associate-+l+ 66.3%

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

      14. *-commutative 66.3%

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

      15. +-commutative 66.3%

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

      16. *-commutative 66.3%

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

      17. associate-+r+ 66.3%

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

      18. +-commutative 66.3%

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

      19. fma-def 66.3%

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

   4. Applied egg-rr 66.3%

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

      1. *-commutative 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-/l* 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in alpha around 0 77.4%

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

   if 2.25e91 < beta < 1.55999999999999997e218

   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 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in i around inf 60.8%

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

   6. Taylor expanded in alpha around 0 60.0%

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

   7. Taylor expanded in i around 0 60.0%

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

   8. Taylor expanded in alpha around 0 60.8%

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

   if 1.55999999999999997e218 < 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 0.0%

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

   5. Taylor expanded in beta around inf 42.0%

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

      1. associate-/l* 43.7%

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

   7. Simplified 43.7%

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

      1. add-sqr-sqrt 43.7%

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

      2. pow2 43.7%

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

      3. +-commutative 43.7%

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

   9. Applied egg-rr 43.7%

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

       1. *-un-lft-identity 43.7%

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

       2. unpow2 43.7%

          \[\leadsto \frac{1 \cdot i}{\color{blue}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}} \cdot \sqrt{\frac{{\beta}^{2}}{i + \alpha}}}} \]

       3. times-frac 43.7%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}} \cdot \frac{i}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}}} \]

       4. sqrt-div 43.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\beta}^{2}}}{\sqrt{i + \alpha}}}} \cdot \frac{i}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}} \]

       5. unpow2 43.7%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\beta \cdot \beta}}}{\sqrt{i + \alpha}}} \cdot \frac{i}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}} \]

       6. sqrt-prod 43.7%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\beta} \cdot \sqrt{\beta}}}{\sqrt{i + \alpha}}} \cdot \frac{i}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}} \]

       7. add-sqr-sqrt 43.7%

          \[\leadsto \frac{1}{\frac{\color{blue}{\beta}}{\sqrt{i + \alpha}}} \cdot \frac{i}{\sqrt{\frac{{\beta}^{2}}{i + \alpha}}} \]

       8. sqrt-div 43.7%

          \[\leadsto \frac{1}{\frac{\beta}{\sqrt{i + \alpha}}} \cdot \frac{i}{\color{blue}{\frac{\sqrt{{\beta}^{2}}}{\sqrt{i + \alpha}}}} \]

       9. unpow2 43.7%

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

       10. sqrt-prod 87.8%

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

       11. add-sqr-sqrt 88.0%

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

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta}{\sqrt{i + \alpha}}} \cdot \frac{i}{\frac{\beta}{\sqrt{i + \alpha}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\beta + i\right)}{\beta + i \cdot 2} \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}{\left(\beta + i\right) + \alpha}}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{\sqrt{i + \alpha}}} \cdot \frac{i}{\frac{\beta}{\sqrt{i + \alpha}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := 0.125 \cdot \frac{\beta}{i}\\ t_4 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t_4 \cdot \left(t_4 + \beta \cdot \alpha\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot \left(\beta + i\right)}{\beta + i \cdot 2} \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}{\left(\beta + i\right) + \alpha}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t_3\right) - t_3\\ \end{array} \end{array} \]

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 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* 0.125 (/ beta i)))
        (t_4 (* i (+ i (+ beta alpha)))))
   (if (<= (/ (/ (* t_4 (+ t_4 (* beta alpha))) t_1) t_2) INFINITY)
     (/
      (*
       (/ (* i (+ beta i)) (+ beta (* i 2.0)))
       (/ i (/ (+ beta (fma i 2.0 alpha)) (+ (+ beta i) alpha))))
      t_2)
     (- (+ 0.0625 t_3) t_3))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = 0.125 * (beta / i);
	double t_4 = i * (i + (beta + alpha));
	double tmp;
	if ((((t_4 * (t_4 + (beta * alpha))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (((i * (beta + i)) / (beta + (i * 2.0))) * (i / ((beta + fma(i, 2.0, alpha)) / ((beta + i) + alpha)))) / t_2;
	} else {
		tmp = (0.0625 + t_3) - t_3;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(0.125 * Float64(beta / i))
	t_4 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(beta * alpha))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(Float64(i * Float64(beta + i)) / Float64(beta + Float64(i * 2.0))) * Float64(i / Float64(Float64(beta + fma(i, 2.0, alpha)) / Float64(Float64(beta + i) + alpha)))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + t_3) - t_3);
	end
	return 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[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + i), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + t$95$3), $MachinePrecision] - t$95$3), $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)) < +inf.0

   1. Initial program 47.4%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-udef 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+l+ 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. *-commutative 99.7%

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

      17. associate-+r+ 99.7%

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

      18. +-commutative 99.7%

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

      19. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in alpha around 0 91.3%

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

   if +inf.0 < (/.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 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 0.0%

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

   5. Taylor expanded in i around inf 76.2%

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

   6. Taylor expanded in alpha around 0 70.9%

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

   7. Taylor expanded in i around 0 70.9%

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

   8. Taylor expanded in alpha around 0 72.3%

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

4. Final simplification 78.8%

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

Alternative 3: 82.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))))
   (if (<= beta 1.56e+218) (- (+ 0.0625 t_0) t_0) (* (/ i beta) (/ i beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 1.56e+218) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.125d0 * (beta / i)
    if (beta <= 1.56d+218) then
        tmp = (0.0625d0 + t_0) - t_0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 1.56e+218) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	tmp = 0
	if beta <= 1.56e+218:
		tmp = (0.0625 + t_0) - t_0
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (beta <= 1.56e+218)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = 0.0;
	if (beta <= 1.56e+218)
		tmp = (0.0625 + t_0) - t_0;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.56e+218], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55999999999999997e218

   1. Initial program 17.8%

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

      1. associate-/l/ 15.6%

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

      2. associate-*l* 15.5%

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

      3. times-frac 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in i around inf 81.2%

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

   6. Taylor expanded in alpha around 0 76.5%

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

   7. Taylor expanded in i around 0 76.5%

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

   8. Taylor expanded in alpha around 0 77.9%

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

   if 1.55999999999999997e218 < 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 0.0%

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

   5. Taylor expanded in beta around inf 42.0%

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

      1. associate-/l* 43.7%

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

   7. Simplified 43.7%

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

   8. Taylor expanded in alpha around 0 43.7%

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

      1. add-sqr-sqrt 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. times-frac 43.7%

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

      4. sqrt-div 43.7%

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

      5. unpow2 43.7%

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

      6. sqrt-prod 43.7%

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

      7. add-sqr-sqrt 43.7%

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

      8. associate-/l* 43.7%

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

      9. add-sqr-sqrt 43.7%

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

      10. sqrt-div 43.7%

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

      11. unpow2 43.7%

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

      12. sqrt-prod 80.1%

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

      13. add-sqr-sqrt 80.1%

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

      14. associate-/l* 80.2%

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

      15. add-sqr-sqrt 80.4%

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

   10. Applied egg-rr 80.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+218) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+218) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.56d+218) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+218) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+218:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+218)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+218)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+218], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55999999999999997e218

   1. Initial program 17.8%

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

      1. associate-/l/ 15.6%

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

      2. associate-*l* 15.5%

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

      3. times-frac 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in i around inf 77.4%

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

   if 1.55999999999999997e218 < 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 0.0%

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

   5. Taylor expanded in beta around inf 42.0%

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

      1. associate-/l* 43.7%

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

   7. Simplified 43.7%

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

   8. Taylor expanded in alpha around 0 43.7%

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

      1. add-sqr-sqrt 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. times-frac 43.7%

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

      4. sqrt-div 43.7%

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

      5. unpow2 43.7%

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

      6. sqrt-prod 43.7%

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

      7. add-sqr-sqrt 43.7%

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

      8. associate-/l* 43.7%

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

      9. add-sqr-sqrt 43.7%

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

      10. sqrt-div 43.7%

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

      11. unpow2 43.7%

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

      12. sqrt-prod 80.1%

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

      13. add-sqr-sqrt 80.1%

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

      14. associate-/l* 80.2%

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

      15. add-sqr-sqrt 80.4%

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

   10. Applied egg-rr 80.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+218}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% 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 16.1%

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

   1. associate-/l/ 14.1%

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

   2. associate-*l* 14.0%

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

   3. times-frac 23.2%

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

3. Simplified 23.2%

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

5. Taylor expanded in i around inf 71.5%

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

6. Final simplification 71.5%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))