Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 84.3%

Time: 21.2s

Alternatives: 7

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + i \cdot 2\\ t_1 := \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ t_2 := i + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{i \cdot \frac{t_2}{\frac{\beta \cdot \beta + \left({t_0}^{2} + 2 \cdot \left(\beta \cdot t_0\right)\right)}{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right)}}}{\left(\mathsf{fma}\left(4, i \cdot \left(\beta + \alpha\right), {\left(\beta + \alpha\right)}^{2}\right) + 4 \cdot \left(i \cdot i\right)\right) + -1}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (* i 2.0)))
        (t_1 (sqrt (* (/ i beta) (/ (+ i alpha) beta))))
        (t_2 (+ i (+ beta alpha))))
   (if (<= beta 5.7e+88)
     0.0625
     (if (<= beta 5.8e+153)
       (/
        (*
         i
         (/
          t_2
          (/
           (+ (* beta beta) (+ (pow t_0 2.0) (* 2.0 (* beta t_0))))
           (fma i t_2 (* beta alpha)))))
        (+
         (+
          (fma 4.0 (* i (+ beta alpha)) (pow (+ beta alpha) 2.0))
          (* 4.0 (* i i)))
         -1.0))
       (if (<= beta 6.4e+154) 0.0625 (* t_1 t_1))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + (i * 2.0);
	double t_1 = sqrt(((i / beta) * ((i + alpha) / beta)));
	double t_2 = i + (beta + alpha);
	double tmp;
	if (beta <= 5.7e+88) {
		tmp = 0.0625;
	} else if (beta <= 5.8e+153) {
		tmp = (i * (t_2 / (((beta * beta) + (pow(t_0, 2.0) + (2.0 * (beta * t_0)))) / fma(i, t_2, (beta * alpha))))) / ((fma(4.0, (i * (beta + alpha)), pow((beta + alpha), 2.0)) + (4.0 * (i * i))) + -1.0);
	} else if (beta <= 6.4e+154) {
		tmp = 0.0625;
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(i * 2.0))
	t_1 = sqrt(Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta)))
	t_2 = Float64(i + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5.7e+88)
		tmp = 0.0625;
	elseif (beta <= 5.8e+153)
		tmp = Float64(Float64(i * Float64(t_2 / Float64(Float64(Float64(beta * beta) + Float64((t_0 ^ 2.0) + Float64(2.0 * Float64(beta * t_0)))) / fma(i, t_2, Float64(beta * alpha))))) / Float64(Float64(fma(4.0, Float64(i * Float64(beta + alpha)), (Float64(beta + alpha) ^ 2.0)) + Float64(4.0 * Float64(i * i))) + -1.0));
	elseif (beta <= 6.4e+154)
		tmp = 0.0625;
	else
		tmp = Float64(t_1 * t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.7e+88], 0.0625, If[LessEqual[beta, 5.8e+153], N[(N[(i * N[(t$95$2 / N[(N[(N[(beta * beta), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(2.0 * N[(beta * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * N[(i * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.4e+154], 0.0625, N[(t$95$1 * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.70000000000000021e88 or 5.80000000000000004e153 < beta < 6.4e154

   1. Initial program 20.4%

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

      1. associate-/l/ 17.8%

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

      2. associate-*l* 17.7%

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

      3. times-frac 25.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in i around inf 85.4%

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

   if 5.70000000000000021e88 < beta < 5.80000000000000004e153

   1. Initial program 19.8%

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

      1. expm1-log1p-u 19.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 19.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 52.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 56.8%

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

      3. associate-*r/ 57.0%

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

      4. +-commutative 57.0%

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

      5. +-commutative 57.0%

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

      6. +-commutative 57.0%

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

      7. +-commutative 57.0%

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

      8. +-commutative 57.0%

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

      9. *-commutative 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in beta around -inf 57.0%

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

      1. unpow2 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in i around 0 57.2%

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

       1. associate-+r+ 57.2%

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

       2. fma-def 57.2%

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

       3. unpow2 57.2%

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

   11. Simplified 57.2%

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

   if 6.4e154 < 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. expm1-log1p-u 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 0.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 8.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 8.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 8.8%

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

      3. associate-*r/ 8.8%

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

      4. +-commutative 8.8%

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

      5. +-commutative 8.8%

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

      6. +-commutative 8.8%

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

      7. +-commutative 8.8%

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

      8. +-commutative 8.8%

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

      9. *-commutative 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in beta around inf 14.8%

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

      1. unpow2 14.8%

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

   8. Simplified 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. times-frac 14.8%

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

      3. times-frac 69.9%

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

   10. Applied egg-rr 69.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)}{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}}}{\left(\mathsf{fma}\left(4, i \cdot \left(\beta + \alpha\right), {\left(\beta + \alpha\right)}^{2}\right) + 4 \cdot \left(i \cdot i\right)\right) + -1}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \cdot \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ \end{array} \]

Alternative 2: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ t_2 := \alpha + i \cdot 2\\ t_3 := i + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{i \cdot \frac{t_3}{\frac{\beta \cdot \beta + \left({t_2}^{2} + 2 \cdot \left(\beta \cdot t_2\right)\right)}{\mathsf{fma}\left(i, t_3, \beta \cdot \alpha\right)}}}{t_0 \cdot t_0 + -1}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (sqrt (* (/ i beta) (/ (+ i alpha) beta))))
        (t_2 (+ alpha (* i 2.0)))
        (t_3 (+ i (+ beta alpha))))
   (if (<= beta 5.7e+88)
     0.0625
     (if (<= beta 7e+152)
       (/
        (*
         i
         (/
          t_3
          (/
           (+ (* beta beta) (+ (pow t_2 2.0) (* 2.0 (* beta t_2))))
           (fma i t_3 (* beta alpha)))))
        (+ (* t_0 t_0) -1.0))
       (if (<= beta 6.4e+154) 0.0625 (* t_1 t_1))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = sqrt(((i / beta) * ((i + alpha) / beta)));
	double t_2 = alpha + (i * 2.0);
	double t_3 = i + (beta + alpha);
	double tmp;
	if (beta <= 5.7e+88) {
		tmp = 0.0625;
	} else if (beta <= 7e+152) {
		tmp = (i * (t_3 / (((beta * beta) + (pow(t_2, 2.0) + (2.0 * (beta * t_2)))) / fma(i, t_3, (beta * alpha))))) / ((t_0 * t_0) + -1.0);
	} else if (beta <= 6.4e+154) {
		tmp = 0.0625;
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = sqrt(Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta)))
	t_2 = Float64(alpha + Float64(i * 2.0))
	t_3 = Float64(i + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5.7e+88)
		tmp = 0.0625;
	elseif (beta <= 7e+152)
		tmp = Float64(Float64(i * Float64(t_3 / Float64(Float64(Float64(beta * beta) + Float64((t_2 ^ 2.0) + Float64(2.0 * Float64(beta * t_2)))) / fma(i, t_3, Float64(beta * alpha))))) / Float64(Float64(t_0 * t_0) + -1.0));
	elseif (beta <= 6.4e+154)
		tmp = 0.0625;
	else
		tmp = Float64(t_1 * t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.7e+88], 0.0625, If[LessEqual[beta, 7e+152], N[(N[(i * N[(t$95$3 / N[(N[(N[(beta * beta), $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(2.0 * N[(beta * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * t$95$3 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.4e+154], 0.0625, N[(t$95$1 * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.70000000000000021e88 or 6.99999999999999963e152 < beta < 6.4e154

   1. Initial program 20.4%

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

      1. associate-/l/ 17.8%

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

      2. associate-*l* 17.7%

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

      3. times-frac 25.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in i around inf 85.4%

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

   if 5.70000000000000021e88 < beta < 6.99999999999999963e152

   1. Initial program 19.8%

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

      1. expm1-log1p-u 19.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 19.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 52.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 56.8%

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

      3. associate-*r/ 57.0%

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

      4. +-commutative 57.0%

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

      5. +-commutative 57.0%

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

      6. +-commutative 57.0%

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

      7. +-commutative 57.0%

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

      8. +-commutative 57.0%

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

      9. *-commutative 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in beta around -inf 57.0%

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

      1. unpow2 57.0%

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

   8. Simplified 57.0%

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

   if 6.4e154 < 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. expm1-log1p-u 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 0.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 8.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 8.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 8.8%

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

      3. associate-*r/ 8.8%

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

      4. +-commutative 8.8%

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

      5. +-commutative 8.8%

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

      6. +-commutative 8.8%

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

      7. +-commutative 8.8%

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

      8. +-commutative 8.8%

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

      9. *-commutative 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in beta around inf 14.8%

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

      1. unpow2 14.8%

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

   8. Simplified 14.8%

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

      1. add-sqr-sqrt 14.8%

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

      2. times-frac 14.8%

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

      3. times-frac 69.9%

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

   10. Applied egg-rr 69.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)}{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \cdot \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \beta \cdot \beta + 4 \cdot \left(i \cdot i\right)\right)}{i \cdot \left(\beta + i\right)}}}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ i beta) (/ (+ i alpha) beta))))
        (t_1 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 4e+88)
     0.0625
     (if (<= beta 1.45e+110)
       (/
        (*
         i
         (/
          (+ i (+ beta alpha))
          (/
           (fma 4.0 (* beta i) (+ (* beta beta) (* 4.0 (* i i))))
           (* i (+ beta i)))))
        (+ (* t_1 t_1) -1.0))
       (if (<= beta 2.9e+121) 0.0625 (* t_0 t_0))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = sqrt(((i / beta) * ((i + alpha) / beta)));
	double t_1 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 4e+88) {
		tmp = 0.0625;
	} else if (beta <= 1.45e+110) {
		tmp = (i * ((i + (beta + alpha)) / (fma(4.0, (beta * i), ((beta * beta) + (4.0 * (i * i)))) / (i * (beta + i))))) / ((t_1 * t_1) + -1.0);
	} else if (beta <= 2.9e+121) {
		tmp = 0.0625;
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = sqrt(Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta)))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4e+88)
		tmp = 0.0625;
	elseif (beta <= 1.45e+110)
		tmp = Float64(Float64(i * Float64(Float64(i + Float64(beta + alpha)) / Float64(fma(4.0, Float64(beta * i), Float64(Float64(beta * beta) + Float64(4.0 * Float64(i * i)))) / Float64(i * Float64(beta + i))))) / Float64(Float64(t_1 * t_1) + -1.0));
	elseif (beta <= 2.9e+121)
		tmp = 0.0625;
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[Sqrt[N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+88], 0.0625, If[LessEqual[beta, 1.45e+110], N[(N[(i * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.9e+121], 0.0625, N[(t$95$0 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.99999999999999984e88 or 1.45e110 < beta < 2.8999999999999999e121

   1. Initial program 20.3%

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

      1. associate-/l/ 17.7%

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

      2. associate-*l* 17.6%

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

      3. times-frac 25.5%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in i around inf 85.5%

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

   if 3.99999999999999984e88 < beta < 1.45e110

   1. Initial program 40.6%

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

      1. expm1-log1p-u 39.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 39.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 75.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 75.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 80.2%

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

      3. associate-*r/ 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. +-commutative 80.5%

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

      7. +-commutative 80.5%

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

      8. +-commutative 80.5%

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

      9. *-commutative 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in beta around -inf 80.5%

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

      1. unpow2 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in alpha around 0 80.5%

      \[\leadsto \frac{i \cdot \frac{\left(\beta + \alpha\right) + i}{\color{blue}{\frac{4 \cdot \left(\beta \cdot i\right) + \left({\beta}^{2} + 4 \cdot {i}^{2}\right)}{i \cdot \left(\beta + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   10. Step-by-step derivation

       1. fma-def 80.5%

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

       2. unpow2 80.5%

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

       3. unpow2 80.5%

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

   11. Simplified 80.5%

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

   if 2.8999999999999999e121 < beta

   1. Initial program 3.1%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 17.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 17.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 18.3%

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

      3. associate-*r/ 18.4%

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

      4. +-commutative 18.4%

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

      5. +-commutative 18.4%

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

      6. +-commutative 18.4%

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

      7. +-commutative 18.4%

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

      8. +-commutative 18.4%

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

      9. *-commutative 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in beta around inf 20.8%

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

      1. unpow2 20.8%

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

   8. Simplified 20.8%

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

      1. add-sqr-sqrt 20.8%

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

      2. times-frac 20.8%

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

      3. times-frac 64.1%

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

   10. Applied egg-rr 64.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \beta \cdot \beta + 4 \cdot \left(i \cdot i\right)\right)}{i \cdot \left(\beta + i\right)}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \cdot \sqrt{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}}\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \beta \cdot \beta + 4 \cdot \left(i \cdot i\right)\right)}{i \cdot \left(\beta + i\right)}}}{t_0 \cdot t_0 + -1}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 6e+88)
     0.0625
     (if (<= beta 3.5e+110)
       (/
        (*
         i
         (/
          (+ i (+ beta alpha))
          (/
           (fma 4.0 (* beta i) (+ (* beta beta) (* 4.0 (* i i))))
           (* i (+ beta i)))))
        (+ (* t_0 t_0) -1.0))
       (if (<= beta 2.9e+121) 0.0625 (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else if (beta <= 3.5e+110) {
		tmp = (i * ((i + (beta + alpha)) / (fma(4.0, (beta * i), ((beta * beta) + (4.0 * (i * i)))) / (i * (beta + i))))) / ((t_0 * t_0) + -1.0);
	} else if (beta <= 2.9e+121) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	elseif (beta <= 3.5e+110)
		tmp = Float64(Float64(i * Float64(Float64(i + Float64(beta + alpha)) / Float64(fma(4.0, Float64(beta * i), Float64(Float64(beta * beta) + Float64(4.0 * Float64(i * i)))) / Float64(i * Float64(beta + i))))) / Float64(Float64(t_0 * t_0) + -1.0));
	elseif (beta <= 2.9e+121)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6e+88], 0.0625, If[LessEqual[beta, 3.5e+110], N[(N[(i * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] + N[(4.0 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.9e+121], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.00000000000000011e88 or 3.4999999999999999e110 < beta < 2.8999999999999999e121

   1. Initial program 20.3%

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

      1. associate-/l/ 17.7%

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

      2. associate-*l* 17.6%

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

      3. times-frac 25.5%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in i around inf 85.5%

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

   if 6.00000000000000011e88 < beta < 3.4999999999999999e110

   1. Initial program 40.6%

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

      1. expm1-log1p-u 39.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 39.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 75.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 75.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 80.2%

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

      3. associate-*r/ 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. +-commutative 80.5%

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

      7. +-commutative 80.5%

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

      8. +-commutative 80.5%

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

      9. *-commutative 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in beta around -inf 80.5%

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

      1. unpow2 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in alpha around 0 80.5%

      \[\leadsto \frac{i \cdot \frac{\left(\beta + \alpha\right) + i}{\color{blue}{\frac{4 \cdot \left(\beta \cdot i\right) + \left({\beta}^{2} + 4 \cdot {i}^{2}\right)}{i \cdot \left(\beta + i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   10. Step-by-step derivation

       1. fma-def 80.5%

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

       2. unpow2 80.5%

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

       3. unpow2 80.5%

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

   11. Simplified 80.5%

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

   if 2.8999999999999999e121 < beta

   1. Initial program 3.1%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 17.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 17.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 18.3%

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

      3. associate-*r/ 18.4%

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

      4. +-commutative 18.4%

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

      5. +-commutative 18.4%

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

      6. +-commutative 18.4%

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

      7. +-commutative 18.4%

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

      8. +-commutative 18.4%

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

      9. *-commutative 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in beta around inf 20.8%

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

      1. unpow2 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. times-frac 64.1%

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

   10. Applied egg-rr 64.1%

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

       1. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(4, \beta \cdot i, \beta \cdot \beta + 4 \cdot \left(i \cdot i\right)\right)}{i \cdot \left(\beta + i\right)}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{i \cdot \left(\beta + i\right)}}}{t_0 \cdot t_0 + -1}\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 5e+88)
     0.0625
     (if (<= beta 1.45e+110)
       (/
        (*
         i
         (/
          (+ i (+ beta alpha))
          (/ (pow (+ beta (* i 2.0)) 2.0) (* i (+ beta i)))))
        (+ (* t_0 t_0) -1.0))
       (if (<= beta 2.8e+121) 0.0625 (* (/ i beta) (/ (+ i alpha) beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 5e+88) {
		tmp = 0.0625;
	} else if (beta <= 1.45e+110) {
		tmp = (i * ((i + (beta + alpha)) / (pow((beta + (i * 2.0)), 2.0) / (i * (beta + i))))) / ((t_0 * t_0) + -1.0);
	} else if (beta <= 2.8e+121) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + (i * 2.0d0)
    if (beta <= 5d+88) then
        tmp = 0.0625d0
    else if (beta <= 1.45d+110) then
        tmp = (i * ((i + (beta + alpha)) / (((beta + (i * 2.0d0)) ** 2.0d0) / (i * (beta + i))))) / ((t_0 * t_0) + (-1.0d0))
    else if (beta <= 2.8d+121) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 5e+88) {
		tmp = 0.0625;
	} else if (beta <= 1.45e+110) {
		tmp = (i * ((i + (beta + alpha)) / (Math.pow((beta + (i * 2.0)), 2.0) / (i * (beta + i))))) / ((t_0 * t_0) + -1.0);
	} else if (beta <= 2.8e+121) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	tmp = 0
	if beta <= 5e+88:
		tmp = 0.0625
	elif beta <= 1.45e+110:
		tmp = (i * ((i + (beta + alpha)) / (math.pow((beta + (i * 2.0)), 2.0) / (i * (beta + i))))) / ((t_0 * t_0) + -1.0)
	elif beta <= 2.8e+121:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 5e+88)
		tmp = 0.0625;
	elseif (beta <= 1.45e+110)
		tmp = Float64(Float64(i * Float64(Float64(i + Float64(beta + alpha)) / Float64((Float64(beta + Float64(i * 2.0)) ^ 2.0) / Float64(i * Float64(beta + i))))) / Float64(Float64(t_0 * t_0) + -1.0));
	elseif (beta <= 2.8e+121)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	tmp = 0.0;
	if (beta <= 5e+88)
		tmp = 0.0625;
	elseif (beta <= 1.45e+110)
		tmp = (i * ((i + (beta + alpha)) / (((beta + (i * 2.0)) ^ 2.0) / (i * (beta + i))))) / ((t_0 * t_0) + -1.0);
	elseif (beta <= 2.8e+121)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+88], 0.0625, If[LessEqual[beta, 1.45e+110], N[(N[(i * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.8e+121], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.99999999999999997e88 or 1.45e110 < beta < 2.80000000000000006e121

   1. Initial program 20.3%

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

      1. associate-/l/ 17.7%

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

      2. associate-*l* 17.6%

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

      3. times-frac 25.5%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in i around inf 85.5%

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

   if 4.99999999999999997e88 < beta < 1.45e110

   1. Initial program 40.6%

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

      1. expm1-log1p-u 39.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 39.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 75.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 75.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 80.2%

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

      3. associate-*r/ 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. +-commutative 80.5%

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

      7. +-commutative 80.5%

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

      8. +-commutative 80.5%

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

      9. *-commutative 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in alpha around 0 80.5%

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

   if 2.80000000000000006e121 < beta

   1. Initial program 3.1%

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

      1. expm1-log1p-u 3.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 3.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 17.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 17.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 18.3%

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

      3. associate-*r/ 18.4%

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

      4. +-commutative 18.4%

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

      5. +-commutative 18.4%

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

      6. +-commutative 18.4%

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

      7. +-commutative 18.4%

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

      8. +-commutative 18.4%

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

      9. *-commutative 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in beta around inf 20.8%

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

      1. unpow2 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. times-frac 64.1%

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

   10. Applied egg-rr 64.1%

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

       1. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{i \cdot \left(\beta + i\right)}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+121}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6: 83.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+88) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6d+88) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+88:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+88], 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 < 6.00000000000000011e88

   1. Initial program 20.5%

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

      1. associate-/l/ 17.9%

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

      2. associate-*l* 17.8%

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

      3. times-frac 25.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in i around inf 85.4%

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

   if 6.00000000000000011e88 < beta

   1. Initial program 5.5%

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

      1. expm1-log1p-u 5.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 5.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 20.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 20.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 21.9%

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

      3. associate-*r/ 22.0%

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

      4. +-commutative 22.0%

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

      5. +-commutative 22.0%

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

      6. +-commutative 22.0%

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

      7. +-commutative 22.0%

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

      8. +-commutative 22.0%

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

      9. *-commutative 22.0%

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

   5. Simplified 22.0%

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

   6. Taylor expanded in beta around inf 22.6%

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

      1. unpow2 22.6%

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

   8. Simplified 22.6%

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

      1. *-un-lft-identity 22.6%

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

      2. times-frac 61.8%

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

   10. Applied egg-rr 61.8%

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

       1. *-lft-identity 61.8%

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

   12. Simplified 61.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 7: 70.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

   1. associate-/l/ 12.6%

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

   2. associate-*l* 12.6%

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

   3. times-frac 20.5%

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

3. Simplified 34.5%

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

4. Taylor expanded in i around inf 68.7%

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

5. Final simplification 68.7%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023216 
(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)))