Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 84.2%

Time: 15.3s

Alternatives: 10

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_1, t_1, -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_1} \cdot \frac{t_0}{t_1}\right)\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ beta alpha))) (t_1 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 7.2e+109)
     0.0625
     (if (<= beta 1.45e+129)
       (*
        (/ i (fma t_1 t_1 -1.0))
        (* (/ (fma i t_0 (* beta alpha)) t_1) (/ t_0 t_1)))
       (if (<= beta 3.5e+194) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = i + (beta + alpha);
	double t_1 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 7.2e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.45e+129) {
		tmp = (i / fma(t_1, t_1, -1.0)) * ((fma(i, t_0, (beta * alpha)) / t_1) * (t_0 / t_1));
	} else if (beta <= 3.5e+194) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(i + Float64(beta + alpha))
	t_1 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 7.2e+109)
		tmp = 0.0625;
	elseif (beta <= 1.45e+129)
		tmp = Float64(Float64(i / fma(t_1, t_1, -1.0)) * Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / t_1) * Float64(t_0 / t_1)));
	elseif (beta <= 3.5e+194)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	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[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.2e+109], 0.0625, If[LessEqual[beta, 1.45e+129], N[(N[(i / N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.5e+194], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 7.2e109 or 1.45000000000000001e129 < beta < 3.4999999999999997e194

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 7.2e109 < beta < 1.45000000000000001e129

   1. Initial program 2.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 34.4%

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

   3. Simplified 77.8%

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

   if 3.4999999999999997e194 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 52.7%

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

      2. associate-/r/ 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in beta around 0 38.9%

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

       1. *-commutative 38.9%

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

       2. associate-/l* 41.0%

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

       3. unpow2 41.0%

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

       4. associate-*r/ 52.7%

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

       5. associate-/r* 79.7%

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

       6. +-commutative 79.7%

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

   13. Simplified 79.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 2: 83.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.38 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, i + \alpha, \frac{i \cdot i}{\frac{\beta}{i + \alpha}}\right) + \frac{{\left(i + \alpha\right)}^{2}}{\frac{\beta}{i}}\right) - \frac{i}{\frac{\beta}{\left(i + \alpha\right) \cdot \mathsf{fma}\left(4, i, \alpha \cdot 2\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0))))
   (if (<= beta 9.5e+109)
     0.0625
     (if (<= beta 1.38e+129)
       (/
        (-
         (+
          (fma i (+ i alpha) (/ (* i i) (/ beta (+ i alpha))))
          (/ (pow (+ i alpha) 2.0) (/ beta i)))
         (/ i (/ beta (* (+ i alpha) (fma 4.0 i (* alpha 2.0))))))
        (+ -1.0 (* t_0 t_0)))
       (if (<= beta 3.5e+194) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double tmp;
	if (beta <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.38e+129) {
		tmp = ((fma(i, (i + alpha), ((i * i) / (beta / (i + alpha)))) + (pow((i + alpha), 2.0) / (beta / i))) - (i / (beta / ((i + alpha) * fma(4.0, i, (alpha * 2.0)))))) / (-1.0 + (t_0 * t_0));
	} else if (beta <= 3.5e+194) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	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 <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.38e+129)
		tmp = Float64(Float64(Float64(fma(i, Float64(i + alpha), Float64(Float64(i * i) / Float64(beta / Float64(i + alpha)))) + Float64((Float64(i + alpha) ^ 2.0) / Float64(beta / i))) - Float64(i / Float64(beta / Float64(Float64(i + alpha) * fma(4.0, i, Float64(alpha * 2.0)))))) / Float64(-1.0 + Float64(t_0 * t_0)));
	elseif (beta <= 3.5e+194)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	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, 9.5e+109], 0.0625, If[LessEqual[beta, 1.38e+129], N[(N[(N[(N[(i * N[(i + alpha), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] / N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(i + alpha), $MachinePrecision], 2.0], $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / N[(beta / N[(N[(i + alpha), $MachinePrecision] * N[(4.0 * i + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.5e+194], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9.49999999999999972e109 or 1.3800000000000001e129 < beta < 3.4999999999999997e194

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 9.49999999999999972e109 < beta < 1.3800000000000001e129

   1. Initial program 2.0%

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

   2. Taylor expanded in beta around inf 55.2%

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

      1. +-commutative 55.2%

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

      2. fma-def 55.2%

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

      3. +-commutative 55.2%

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

      4. *-commutative 55.2%

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

      5. associate-/l* 55.2%

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

      6. unpow2 55.2%

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

      7. +-commutative 55.2%

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

      8. associate-/l* 55.4%

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

      9. +-commutative 55.4%

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

      10. associate-/l* 77.3%

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

      11. +-commutative 77.3%

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

      12. fma-def 77.3%

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

      13. *-commutative 77.3%

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

   4. Simplified 77.3%

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

   if 3.4999999999999997e194 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 52.7%

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

      2. associate-/r/ 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in beta around 0 38.9%

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

       1. *-commutative 38.9%

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

       2. associate-/l* 41.0%

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

       3. unpow2 41.0%

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

       4. associate-*r/ 52.7%

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

       5. associate-/r* 79.7%

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

       6. +-commutative 79.7%

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

   13. Simplified 79.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.38 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, i + \alpha, \frac{i \cdot i}{\frac{\beta}{i + \alpha}}\right) + \frac{{\left(i + \alpha\right)}^{2}}{\frac{\beta}{i}}\right) - \frac{i}{\frac{\beta}{\left(i + \alpha\right) \cdot \mathsf{fma}\left(4, i, \alpha \cdot 2\right)}}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 3: 83.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.5e+109)
   0.0625
   (if (<= beta 1.4e+129)
     (/ (+ i alpha) (/ (* beta beta) i))
     (if (<= beta 1.65e+194) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+129) {
		tmp = (i + alpha) / ((beta * beta) / i);
	} else if (beta <= 1.65e+194) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 9.5d+109) then
        tmp = 0.0625d0
    else if (beta <= 1.4d+129) then
        tmp = (i + alpha) / ((beta * beta) / i)
    else if (beta <= 1.65d+194) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+129) {
		tmp = (i + alpha) / ((beta * beta) / i);
	} else if (beta <= 1.65e+194) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.5e+109:
		tmp = 0.0625
	elif beta <= 1.4e+129:
		tmp = (i + alpha) / ((beta * beta) / i)
	elif beta <= 1.65e+194:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.4e+129)
		tmp = Float64(Float64(i + alpha) / Float64(Float64(beta * beta) / i));
	elseif (beta <= 1.65e+194)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.4e+129)
		tmp = (i + alpha) / ((beta * beta) / i);
	elseif (beta <= 1.65e+194)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 9.5e+109], 0.0625, If[LessEqual[beta, 1.4e+129], N[(N[(i + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.65e+194], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9.49999999999999972e109 or 1.39999999999999987e129 < beta < 1.64999999999999992e194

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 9.49999999999999972e109 < beta < 1.39999999999999987e129

   1. Initial program 2.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 34.4%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in beta around inf 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 75.9%

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

      3. +-commutative 75.9%

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

      4. unpow2 75.9%

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

   6. Simplified 75.9%

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

   if 1.64999999999999992e194 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 52.7%

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

      2. associate-/r/ 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in beta around 0 38.9%

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

       1. *-commutative 38.9%

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

       2. associate-/l* 41.0%

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

       3. unpow2 41.0%

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

       4. associate-*r/ 52.7%

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

       5. associate-/r* 79.7%

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

       6. +-commutative 79.7%

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

   13. Simplified 79.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 4: 82.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.38 \cdot 10^{+129}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+197}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.5e+109)
   0.0625
   (if (<= beta 1.38e+129)
     (* (+ i alpha) (/ (/ i beta) beta))
     (if (<= beta 2.4e+197) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.38e+129) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 2.4e+197) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 9.5d+109) then
        tmp = 0.0625d0
    else if (beta <= 1.38d+129) then
        tmp = (i + alpha) * ((i / beta) / beta)
    else if (beta <= 2.4d+197) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / 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 <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.38e+129) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 2.4e+197) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.5e+109:
		tmp = 0.0625
	elif beta <= 1.38e+129:
		tmp = (i + alpha) * ((i / beta) / beta)
	elif beta <= 2.4e+197:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.38e+129)
		tmp = Float64(Float64(i + alpha) * Float64(Float64(i / beta) / beta));
	elseif (beta <= 2.4e+197)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.38e+129)
		tmp = (i + alpha) * ((i / beta) / beta);
	elseif (beta <= 2.4e+197)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / 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, 9.5e+109], 0.0625, If[LessEqual[beta, 1.38e+129], N[(N[(i + alpha), $MachinePrecision] * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.4e+197], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9.49999999999999972e109 or 1.3800000000000001e129 < beta < 2.3999999999999999e197

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 9.49999999999999972e109 < beta < 1.3800000000000001e129

   1. Initial program 2.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 34.4%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in beta around inf 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 75.9%

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

      3. +-commutative 75.9%

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

      4. unpow2 75.9%

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

   6. Simplified 75.9%

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

      1. div-inv 75.9%

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

      2. clear-num 75.7%

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

      3. associate-/r* 75.7%

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

   8. Applied egg-rr 75.7%

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

   if 2.3999999999999999e197 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

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

   9. Taylor expanded in alpha around 0 39.4%

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

       1. unpow2 39.4%

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

       2. unpow2 39.4%

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

       3. times-frac 74.4%

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

   11. Simplified 74.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.38 \cdot 10^{+129}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+197}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5: 82.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.5e+109)
   0.0625
   (if (<= beta 1.5e+129)
     (/ (+ i alpha) (* beta (/ beta i)))
     (if (<= beta 4.3e+196) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.5e+129) {
		tmp = (i + alpha) / (beta * (beta / i));
	} else if (beta <= 4.3e+196) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 9.5d+109) then
        tmp = 0.0625d0
    else if (beta <= 1.5d+129) then
        tmp = (i + alpha) / (beta * (beta / i))
    else if (beta <= 4.3d+196) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / 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 <= 9.5e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.5e+129) {
		tmp = (i + alpha) / (beta * (beta / i));
	} else if (beta <= 4.3e+196) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.5e+109:
		tmp = 0.0625
	elif beta <= 1.5e+129:
		tmp = (i + alpha) / (beta * (beta / i))
	elif beta <= 4.3e+196:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.5e+129)
		tmp = Float64(Float64(i + alpha) / Float64(beta * Float64(beta / i)));
	elseif (beta <= 4.3e+196)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.5e+109)
		tmp = 0.0625;
	elseif (beta <= 1.5e+129)
		tmp = (i + alpha) / (beta * (beta / i));
	elseif (beta <= 4.3e+196)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / 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, 9.5e+109], 0.0625, If[LessEqual[beta, 1.5e+129], N[(N[(i + alpha), $MachinePrecision] / N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.3e+196], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9.49999999999999972e109 or 1.50000000000000015e129 < beta < 4.30000000000000012e196

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 9.49999999999999972e109 < beta < 1.50000000000000015e129

   1. Initial program 2.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 34.4%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in beta around inf 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 75.9%

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

      3. +-commutative 75.9%

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

      4. unpow2 75.9%

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

   6. Simplified 75.9%

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

      1. *-un-lft-identity 75.9%

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

      2. associate-/l* 75.9%

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

   8. Applied egg-rr 75.9%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 75.9%

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

      2. associate-/r/ 75.7%

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

   10. Simplified 75.7%

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

   if 4.30000000000000012e196 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

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

   9. Taylor expanded in alpha around 0 39.4%

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

       1. unpow2 39.4%

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

       2. unpow2 39.4%

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

       3. times-frac 74.4%

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

   11. Simplified 74.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6: 82.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9e+109)
   0.0625
   (if (<= beta 1.4e+129)
     (/ (+ i alpha) (/ (* beta beta) i))
     (if (<= beta 5.2e+194) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+129) {
		tmp = (i + alpha) / ((beta * beta) / i);
	} else if (beta <= 5.2e+194) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 9d+109) then
        tmp = 0.0625d0
    else if (beta <= 1.4d+129) then
        tmp = (i + alpha) / ((beta * beta) / i)
    else if (beta <= 5.2d+194) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / 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 <= 9e+109) {
		tmp = 0.0625;
	} else if (beta <= 1.4e+129) {
		tmp = (i + alpha) / ((beta * beta) / i);
	} else if (beta <= 5.2e+194) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9e+109:
		tmp = 0.0625
	elif beta <= 1.4e+129:
		tmp = (i + alpha) / ((beta * beta) / i)
	elif beta <= 5.2e+194:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9e+109)
		tmp = 0.0625;
	elseif (beta <= 1.4e+129)
		tmp = Float64(Float64(i + alpha) / Float64(Float64(beta * beta) / i));
	elseif (beta <= 5.2e+194)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9e+109)
		tmp = 0.0625;
	elseif (beta <= 1.4e+129)
		tmp = (i + alpha) / ((beta * beta) / i);
	elseif (beta <= 5.2e+194)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / 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, 9e+109], 0.0625, If[LessEqual[beta, 1.4e+129], N[(N[(i + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5.2e+194], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.9999999999999992e109 or 1.39999999999999987e129 < beta < 5.1999999999999998e194

   1. Initial program 17.6%

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

      1. associate-/l/ 17.1%

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

      2. associate-*l* 17.0%

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

      3. times-frac 22.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 8.9999999999999992e109 < beta < 1.39999999999999987e129

   1. Initial program 2.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 34.4%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in beta around inf 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-/l* 75.9%

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

      3. +-commutative 75.9%

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

      4. unpow2 75.9%

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

   6. Simplified 75.9%

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

   if 5.1999999999999998e194 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

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

   9. Taylor expanded in alpha around 0 39.4%

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

       1. unpow2 39.4%

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

       2. unpow2 39.4%

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

       3. times-frac 74.4%

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

   11. Simplified 74.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+109}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{i + \alpha}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7: 75.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+199) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+199) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (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 <= 2d+199) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (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 <= 2e+199) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+199:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+199)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(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 <= 2e+199)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (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, 2e+199], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000019e199

   1. Initial program 17.0%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 22.5%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in i around inf 78.3%

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

   if 2.00000000000000019e199 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

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

   9. Taylor expanded in alpha around inf 40.4%

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

       1. *-commutative 40.4%

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

       2. unpow2 40.4%

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

       3. times-frac 46.5%

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

   11. Simplified 46.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 8: 83.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+193}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.2e+193) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+193) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 9.2d+193) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / 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 <= 9.2e+193) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.2e+193:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+193)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.2e+193)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / 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, 9.2e+193], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.20000000000000053e193

   1. Initial program 17.0%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 22.5%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in i around inf 78.3%

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

   if 9.20000000000000053e193 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in beta around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 41.0%

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

      3. +-commutative 41.0%

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

      4. unpow2 41.0%

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

   6. Simplified 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. associate-/l* 52.7%

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

   8. Applied egg-rr 52.7%

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

   9. Taylor expanded in alpha around 0 39.4%

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

       1. unpow2 39.4%

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

       2. unpow2 39.4%

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

       3. times-frac 74.4%

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

   11. Simplified 74.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+193}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9: 74.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+199) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+199) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2d+199) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+199) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+199:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+199)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+199)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+199], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000019e199

   1. Initial program 17.0%

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

      1. associate-/l/ 16.4%

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

      2. associate-*l* 16.3%

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

      3. times-frac 22.5%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in i around inf 78.3%

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

   if 2.00000000000000019e199 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in i around inf 50.1%

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

      1. cancel-sign-sub-inv 50.1%

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

      2. distribute-lft-out 50.1%

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

      3. metadata-eval 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in i around 0 41.0%

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

      1. distribute-rgt-out 41.0%

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

      2. metadata-eval 41.0%

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

      3. mul0-rgt 41.0%

         \[\leadsto \frac{\color{blue}{0}}{i} \]

   9. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 10: 71.0% 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 14.9%

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

   1. associate-/l/ 14.4%

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

   2. associate-*l* 14.3%

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

   3. times-frac 19.7%

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

3. Simplified 35.9%

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

4. Taylor expanded in i around inf 70.4%

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

5. Final simplification 70.4%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023244 
(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)))