Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 85.4%

Time: 20.3s

Alternatives: 10

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.4% 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: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + \left(i + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \beta \cdot \left(i + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+147}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{i \cdot t\_0}{t\_1}}{t\_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_0, \beta \cdot \alpha\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{t\_2} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (+ i alpha)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (* beta (+ i alpha))))
   (if (<= beta 1.15e+147)
     0.0625
     (if (<= beta 2e+157)
       (*
        (/ (/ (* i t_0) t_1) (+ t_1 1.0))
        (/ (/ (fma i t_0 (* beta alpha)) t_1) (+ t_1 -1.0)))
       (if (<= beta 5e+181)
         (-
          (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
          (* 0.125 (/ (+ beta alpha) i)))
         (*
          (/ i beta)
          (/
           1.0
           (*
            beta
            (-
             (+
              (* 2.0 (/ alpha t_2))
              (+ (* 4.0 (/ 1.0 beta)) (/ 1.0 (+ i alpha))))
             (/ i t_2))))))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = beta * (i + alpha);
	double tmp;
	if (beta <= 1.15e+147) {
		tmp = 0.0625;
	} else if (beta <= 2e+157) {
		tmp = (((i * t_0) / t_1) / (t_1 + 1.0)) * ((fma(i, t_0, (beta * alpha)) / t_1) / (t_1 + -1.0));
	} else if (beta <= 5e+181) {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_2)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_2))));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(beta * Float64(i + alpha))
	tmp = 0.0
	if (beta <= 1.15e+147)
		tmp = 0.0625;
	elseif (beta <= 2e+157)
		tmp = Float64(Float64(Float64(Float64(i * t_0) / t_1) / Float64(t_1 + 1.0)) * Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / t_1) / Float64(t_1 + -1.0)));
	elseif (beta <= 5e+181)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(1.0 / Float64(beta * Float64(Float64(Float64(2.0 * Float64(alpha / t_2)) + Float64(Float64(4.0 * Float64(1.0 / beta)) + Float64(1.0 / Float64(i + alpha)))) - Float64(i / t_2)))));
	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[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta * N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.15e+147], 0.0625, If[LessEqual[beta, 2e+157], N[(N[(N[(N[(i * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5e+181], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(1.0 / N[(beta * N[(N[(N[(2.0 * N[(alpha / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.15e147

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 1.15e147 < beta < 1.99999999999999997e157

   1. Initial program 0.5%

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

      1. associate-/l/ 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. +-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. distribute-rgt-in 0.0%

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

      5. +-commutative 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. fma-define 0.0%

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

      8. +-commutative 0.0%

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

      9. +-commutative 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 41.1%

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

   if 1.99999999999999997e157 < beta < 5.0000000000000003e181

   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. times-frac 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in i around inf 71.9%

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

   if 5.0000000000000003e181 < 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. times-frac 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in beta around inf 10.8%

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

      1. clear-num 10.8%

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

      2. inv-pow 10.8%

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

      3. +-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. +-commutative 10.8%

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

   7. Applied egg-rr 10.8%

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

      1. unpow-1 10.8%

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

      2. associate-/r/ 10.8%

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

      3. +-commutative 10.8%

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

      4. associate-+r+ 10.8%

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

      5. +-commutative 10.8%

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

      6. +-commutative 10.8%

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

      7. *-commutative 10.8%

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

      8. +-commutative 10.8%

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

   9. Simplified 10.8%

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

   10. Taylor expanded in beta around inf 80.3%

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

   11. Taylor expanded in i around inf 80.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+147}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{\beta \cdot \left(i + \alpha\right)} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{\beta \cdot \left(i + \alpha\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta \cdot \left(i + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \alpha + i \cdot 2\\ t_3 := 2 \cdot \frac{t\_2}{i}\\ \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{t\_1}}{t\_1 \cdot \left(\beta \cdot \left(\frac{\left(t\_3 + \frac{\frac{-1 + {t\_2}^{2}}{i} + \left(i + \alpha\right) \cdot \left(\left(1 + \frac{\alpha}{i}\right) - t\_3\right)}{\beta}\right) + \left(-1 - \frac{\alpha}{i}\right)}{\beta} + \frac{1}{i}\right)\right)}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+184}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{t\_0} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* beta (+ i alpha)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (+ alpha (* i 2.0)))
        (t_3 (* 2.0 (/ t_2 i))))
   (if (<= beta 8.5e+146)
     0.0625
     (if (<= beta 1.75e+157)
       (/
        (/ (fma i (+ beta (+ i alpha)) (* beta alpha)) t_1)
        (*
         t_1
         (*
          beta
          (+
           (/
            (+
             (+
              t_3
              (/
               (+
                (/ (+ -1.0 (pow t_2 2.0)) i)
                (* (+ i alpha) (- (+ 1.0 (/ alpha i)) t_3)))
               beta))
             (- -1.0 (/ alpha i)))
            beta)
           (/ 1.0 i)))))
       (if (<= beta 3.4e+184)
         (-
          (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
          (* 0.125 (/ (+ beta alpha) i)))
         (*
          (/ i beta)
          (/
           1.0
           (*
            beta
            (-
             (+
              (* 2.0 (/ alpha t_0))
              (+ (* 4.0 (/ 1.0 beta)) (/ 1.0 (+ i alpha))))
             (/ i t_0))))))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta * (i + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = alpha + (i * 2.0);
	double t_3 = 2.0 * (t_2 / i);
	double tmp;
	if (beta <= 8.5e+146) {
		tmp = 0.0625;
	} else if (beta <= 1.75e+157) {
		tmp = (fma(i, (beta + (i + alpha)), (beta * alpha)) / t_1) / (t_1 * (beta * ((((t_3 + ((((-1.0 + pow(t_2, 2.0)) / i) + ((i + alpha) * ((1.0 + (alpha / i)) - t_3))) / beta)) + (-1.0 - (alpha / i))) / beta) + (1.0 / i))));
	} else if (beta <= 3.4e+184) {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_0)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_0))));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta * Float64(i + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(alpha + Float64(i * 2.0))
	t_3 = Float64(2.0 * Float64(t_2 / i))
	tmp = 0.0
	if (beta <= 8.5e+146)
		tmp = 0.0625;
	elseif (beta <= 1.75e+157)
		tmp = Float64(Float64(fma(i, Float64(beta + Float64(i + alpha)), Float64(beta * alpha)) / t_1) / Float64(t_1 * Float64(beta * Float64(Float64(Float64(Float64(t_3 + Float64(Float64(Float64(Float64(-1.0 + (t_2 ^ 2.0)) / i) + Float64(Float64(i + alpha) * Float64(Float64(1.0 + Float64(alpha / i)) - t_3))) / beta)) + Float64(-1.0 - Float64(alpha / i))) / beta) + Float64(1.0 / i)))));
	elseif (beta <= 3.4e+184)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(1.0 / Float64(beta * Float64(Float64(Float64(2.0 * Float64(alpha / t_0)) + Float64(Float64(4.0 * Float64(1.0 / beta)) + Float64(1.0 / Float64(i + alpha)))) - Float64(i / t_0)))));
	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[(beta * N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.5e+146], 0.0625, If[LessEqual[beta, 1.75e+157], N[(N[(N[(i * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * N[(beta * N[(N[(N[(N[(t$95$3 + N[(N[(N[(N[(-1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(N[(i + alpha), $MachinePrecision] * N[(N[(1.0 + N[(alpha / i), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(alpha / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.4e+184], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(1.0 / N[(beta * N[(N[(N[(2.0 * N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 8.5e146

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 8.5e146 < beta < 1.75000000000000001e157

   1. Initial program 0.5%

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

      1. associate-/l/ 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. times-frac 20.6%

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

   3. Simplified 20.6%

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

      1. clear-num 20.6%

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

      2. frac-times 20.6%

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

      3. *-un-lft-identity 20.6%

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

      4. *-commutative 20.6%

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

      5. +-commutative 20.6%

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

      6. fma-undefine 20.6%

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

      7. pow2 20.6%

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

      8. +-commutative 20.6%

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

   6. Applied egg-rr 20.6%

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

   7. Taylor expanded in beta around -inf 39.7%

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

   if 1.75000000000000001e157 < beta < 3.4000000000000002e184

   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. times-frac 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in i around inf 71.9%

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

   if 3.4000000000000002e184 < 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. times-frac 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in beta around inf 10.8%

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

      1. clear-num 10.8%

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

      2. inv-pow 10.8%

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

      3. +-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. +-commutative 10.8%

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

   7. Applied egg-rr 10.8%

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

      1. unpow-1 10.8%

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

      2. associate-/r/ 10.8%

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

      3. +-commutative 10.8%

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

      4. associate-+r+ 10.8%

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

      5. +-commutative 10.8%

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

      6. +-commutative 10.8%

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

      7. *-commutative 10.8%

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

      8. +-commutative 10.8%

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

   9. Simplified 10.8%

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

   10. Taylor expanded in beta around inf 80.3%

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

   11. Taylor expanded in i around inf 80.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) \cdot \left(\beta \cdot \left(\frac{\left(2 \cdot \frac{\alpha + i \cdot 2}{i} + \frac{\frac{-1 + {\left(\alpha + i \cdot 2\right)}^{2}}{i} + \left(i + \alpha\right) \cdot \left(\left(1 + \frac{\alpha}{i}\right) - 2 \cdot \frac{\alpha + i \cdot 2}{i}\right)}{\beta}\right) + \left(-1 - \frac{\alpha}{i}\right)}{\beta} + \frac{1}{i}\right)\right)}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+184}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{\beta \cdot \left(i + \alpha\right)} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{\beta \cdot \left(i + \alpha\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \beta \cdot \left(i + \alpha\right)\\ t_2 := \frac{\beta}{i \cdot \alpha}\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+145}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{t\_0}}{t\_0 \cdot \left(\alpha \cdot \left(\left(2 \cdot t\_2 + \left(\frac{1}{i} + 3 \cdot \frac{1}{\alpha}\right)\right) - t\_2\right)\right)}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+181}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{t\_1} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{t\_1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha)))
        (t_1 (* beta (+ i alpha)))
        (t_2 (/ beta (* i alpha))))
   (if (<= beta 2.2e+145)
     0.0625
     (if (<= beta 1.75e+157)
       (/
        (/ (fma i (+ beta (+ i alpha)) (* beta alpha)) t_0)
        (*
         t_0
         (*
          alpha
          (- (+ (* 2.0 t_2) (+ (/ 1.0 i) (* 3.0 (/ 1.0 alpha)))) t_2))))
       (if (<= beta 5.2e+181)
         (-
          (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
          (* 0.125 (/ (+ beta alpha) i)))
         (*
          (/ i beta)
          (/
           1.0
           (*
            beta
            (-
             (+
              (* 2.0 (/ alpha t_1))
              (+ (* 4.0 (/ 1.0 beta)) (/ 1.0 (+ i alpha))))
             (/ i t_1))))))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = beta * (i + alpha);
	double t_2 = beta / (i * alpha);
	double tmp;
	if (beta <= 2.2e+145) {
		tmp = 0.0625;
	} else if (beta <= 1.75e+157) {
		tmp = (fma(i, (beta + (i + alpha)), (beta * alpha)) / t_0) / (t_0 * (alpha * (((2.0 * t_2) + ((1.0 / i) + (3.0 * (1.0 / alpha)))) - t_2)));
	} else if (beta <= 5.2e+181) {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_1)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_1))));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(beta * Float64(i + alpha))
	t_2 = Float64(beta / Float64(i * alpha))
	tmp = 0.0
	if (beta <= 2.2e+145)
		tmp = 0.0625;
	elseif (beta <= 1.75e+157)
		tmp = Float64(Float64(fma(i, Float64(beta + Float64(i + alpha)), Float64(beta * alpha)) / t_0) / Float64(t_0 * Float64(alpha * Float64(Float64(Float64(2.0 * t_2) + Float64(Float64(1.0 / i) + Float64(3.0 * Float64(1.0 / alpha)))) - t_2))));
	elseif (beta <= 5.2e+181)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(1.0 / Float64(beta * Float64(Float64(Float64(2.0 * Float64(alpha / t_1)) + Float64(Float64(4.0 * Float64(1.0 / beta)) + Float64(1.0 / Float64(i + alpha)))) - 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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta * N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta / N[(i * alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2e+145], 0.0625, If[LessEqual[beta, 1.75e+157], N[(N[(N[(i * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha * N[(N[(N[(2.0 * t$95$2), $MachinePrecision] + N[(N[(1.0 / i), $MachinePrecision] + N[(3.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5.2e+181], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(1.0 / N[(beta * N[(N[(N[(2.0 * N[(alpha / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.20000000000000009e145

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 2.20000000000000009e145 < beta < 1.75000000000000001e157

   1. Initial program 0.5%

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

      1. associate-/l/ 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. times-frac 20.6%

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

   3. Simplified 20.6%

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

      1. clear-num 20.6%

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

      2. frac-times 20.6%

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

      3. *-un-lft-identity 20.6%

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

      4. *-commutative 20.6%

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

      5. +-commutative 20.6%

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

      6. fma-undefine 20.6%

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

      7. pow2 20.6%

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

      8. +-commutative 20.6%

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

   6. Applied egg-rr 20.6%

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

   7. Taylor expanded in alpha around inf 37.9%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\color{blue}{\left(\alpha \cdot \left(\left(2 \cdot \frac{\beta}{\alpha \cdot i} + \left(3 \cdot \frac{1}{\alpha} + \frac{1}{i}\right)\right) - \frac{\beta}{\alpha \cdot i}\right)\right)} \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \]

   if 1.75000000000000001e157 < beta < 5.2e181

   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. times-frac 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in i around inf 71.9%

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

   if 5.2e181 < 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. times-frac 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in beta around inf 10.8%

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

      1. clear-num 10.8%

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

      2. inv-pow 10.8%

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

      3. +-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. +-commutative 10.8%

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

   7. Applied egg-rr 10.8%

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

      1. unpow-1 10.8%

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

      2. associate-/r/ 10.8%

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

      3. +-commutative 10.8%

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

      4. associate-+r+ 10.8%

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

      5. +-commutative 10.8%

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

      6. +-commutative 10.8%

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

      7. *-commutative 10.8%

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

      8. +-commutative 10.8%

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

   9. Simplified 10.8%

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

   10. Taylor expanded in beta around inf 80.3%

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

   11. Taylor expanded in i around inf 80.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+145}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) \cdot \left(\alpha \cdot \left(\left(2 \cdot \frac{\beta}{i \cdot \alpha} + \left(\frac{1}{i} + 3 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta}{i \cdot \alpha}\right)\right)}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+181}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{\beta \cdot \left(i + \alpha\right)} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{\beta \cdot \left(i + \alpha\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta \cdot \left(i + \alpha\right)\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{t\_0} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* beta (+ i alpha))))
   (if (<= beta 3.5e+146)
     0.0625
     (*
      (/ i beta)
      (/
       1.0
       (*
        beta
        (-
         (+ (* 2.0 (/ alpha t_0)) (+ (* 4.0 (/ 1.0 beta)) (/ 1.0 (+ i alpha))))
         (/ i t_0))))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta * (i + alpha);
	double tmp;
	if (beta <= 3.5e+146) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_0)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_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) :: t_0
    real(8) :: tmp
    t_0 = beta * (i + alpha)
    if (beta <= 3.5d+146) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (1.0d0 / (beta * (((2.0d0 * (alpha / t_0)) + ((4.0d0 * (1.0d0 / beta)) + (1.0d0 / (i + alpha)))) - (i / t_0))))
    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 * (i + alpha);
	double tmp;
	if (beta <= 3.5e+146) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_0)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_0))));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = beta * (i + alpha)
	tmp = 0
	if beta <= 3.5e+146:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_0)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_0))))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta * Float64(i + alpha))
	tmp = 0.0
	if (beta <= 3.5e+146)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(1.0 / Float64(beta * Float64(Float64(Float64(2.0 * Float64(alpha / t_0)) + Float64(Float64(4.0 * Float64(1.0 / beta)) + Float64(1.0 / Float64(i + alpha)))) - Float64(i / t_0)))));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = beta * (i + alpha);
	tmp = 0.0;
	if (beta <= 3.5e+146)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (1.0 / (beta * (((2.0 * (alpha / t_0)) + ((4.0 * (1.0 / beta)) + (1.0 / (i + alpha)))) - (i / t_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_] := Block[{t$95$0 = N[(beta * N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.5e+146], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(1.0 / N[(beta * N[(N[(N[(2.0 * N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5000000000000001e146

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 3.5000000000000001e146 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in beta around inf 18.1%

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

      1. clear-num 18.1%

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

      2. inv-pow 18.1%

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

      3. +-commutative 18.1%

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

      4. *-commutative 18.1%

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

      5. +-commutative 18.1%

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

   7. Applied egg-rr 18.1%

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

      1. unpow-1 18.1%

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

      2. associate-/r/ 18.2%

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

      3. +-commutative 18.2%

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

      4. associate-+r+ 18.2%

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

      5. +-commutative 18.2%

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

      6. +-commutative 18.2%

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

      7. *-commutative 18.2%

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

      8. +-commutative 18.2%

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

   9. Simplified 18.2%

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

   10. Taylor expanded in beta around inf 64.4%

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

   11. Taylor expanded in i around inf 64.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{1}{\beta \cdot \left(\left(2 \cdot \frac{\alpha}{\beta \cdot \left(i + \alpha\right)} + \left(4 \cdot \frac{1}{\beta} + \frac{1}{i + \alpha}\right)\right) - \frac{i}{\beta \cdot \left(i + \alpha\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+183}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \left(\left(i + \alpha\right) \cdot \frac{1}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.1e+183)
   (-
    (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
    (* 0.125 (/ (+ beta alpha) i)))
   (* (/ i beta) (* (+ i alpha) (/ 1.0 beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.1e+183) {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) * (1.0 / 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 <= 2.1d+183) then
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((beta + alpha) / i))
    else
        tmp = (i / beta) * ((i + alpha) * (1.0d0 / 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 <= 2.1e+183) {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = (i / beta) * ((i + alpha) * (1.0 / beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.1e+183:
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i))
	else:
		tmp = (i / beta) * ((i + alpha) * (1.0 / beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.1e+183)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) * Float64(1.0 / 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 <= 2.1e+183)
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((beta + alpha) / i));
	else
		tmp = (i / beta) * ((i + alpha) * (1.0 / 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, 2.1e+183], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.1e183

   1. Initial program 17.6%

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

      1. associate-/l/ 15.8%

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

      2. times-frac 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in i around inf 80.6%

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

   if 2.1e183 < 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. times-frac 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in beta around inf 10.8%

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

   6. Taylor expanded in beta around inf 80.4%

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

      1. +-commutative 80.4%

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

   8. Simplified 80.4%

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

      1. div-inv 80.4%

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

      2. +-commutative 80.4%

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

   10. Applied egg-rr 80.4%

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

       1. +-commutative 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+183}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \left(\left(i + \alpha\right) \cdot \frac{1}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \left(\left(i + \alpha\right) \cdot \frac{1}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.5e+145) 0.0625 (* (/ i beta) (* (+ i alpha) (/ 1.0 beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+145) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) * (1.0 / 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 <= 9.5d+145) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) * (1.0d0 / 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 <= 9.5e+145) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) * (1.0 / beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.5e+145:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) * (1.0 / beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.5e+145)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) * Float64(1.0 / 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 <= 9.5e+145)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) * (1.0 / 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, 9.5e+145], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.49999999999999948e145

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 9.49999999999999948e145 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in beta around inf 18.1%

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

   6. Taylor expanded in beta around inf 64.4%

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

      1. +-commutative 64.4%

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

   8. Simplified 64.4%

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

      1. div-inv 64.4%

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

      2. +-commutative 64.4%

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

   10. Applied egg-rr 64.4%

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

       1. +-commutative 64.4%

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

   12. Simplified 64.4%

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

Alternative 7: 85.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+146) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+146) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.2d+146) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+146) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.2e+146:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+146)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.2e+146)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+146], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.1999999999999997e146

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 7.1999999999999997e146 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in beta around inf 18.1%

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

   6. Taylor expanded in beta around inf 64.4%

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

      1. +-commutative 64.4%

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

   8. Simplified 64.4%

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

Alternative 8: 83.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.25e+146) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.25e+146) {
		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 <= 1.25d+146) 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 <= 1.25e+146) {
		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 <= 1.25e+146:
		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 <= 1.25e+146)
		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 <= 1.25e+146)
		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, 1.25e+146], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e146

   1. Initial program 19.2%

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

      1. associate-/l/ 17.2%

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

      2. times-frac 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in i around inf 77.6%

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

   if 1.25e146 < beta

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in beta around inf 18.1%

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

   6. Taylor expanded in beta around inf 64.4%

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

      1. +-commutative 64.4%

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

   8. Simplified 64.4%

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

   9. Taylor expanded in i around inf 62.5%

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

Alternative 9: 75.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+218}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+218) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+218) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.6d+218) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+218) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.6e+218:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+218)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.6e+218)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.6e+218], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.59999999999999991e218

   1. Initial program 16.8%

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

      1. associate-/l/ 15.1%

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

      2. times-frac 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in i around inf 73.5%

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

   if 3.59999999999999991e218 < 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. times-frac 4.3%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in beta around inf 6.3%

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

   6. Taylor expanded in beta around inf 87.6%

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

      1. +-commutative 87.6%

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

   8. Simplified 87.6%

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

   9. Taylor expanded in i around 0 45.0%

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

Alternative 10: 71.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

   1. associate-/l/ 13.7%

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

   2. times-frac 33.3%

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

3. Simplified 33.3%

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

5. Taylor expanded in i around inf 67.7%

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

Reproduce

herbie shell --seed 2024100 
(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)))