Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 82.2%

Time: 19.9s

Alternatives: 6

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: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \beta + \left(i + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.75 \cdot 10^{+142}:\\ \;\;\;\;\frac{i \cdot t\_1}{\mathsf{fma}\left(t\_0, t\_0, -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_1, \beta \cdot \alpha\right)}{t\_0}}{t\_0}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))) (t_1 (+ beta (+ i alpha))))
   (if (<= beta 2.6e+111)
     0.0625
     (if (<= beta 4.75e+142)
       (*
        (/ (* i t_1) (fma t_0 t_0 -1.0))
        (/ (/ (fma i t_1 (* beta alpha)) t_0) t_0))
       (if (<= beta 1.05e+200)
         0.0625
         (/ (* i (/ (+ i alpha) beta)) (+ beta alpha)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = beta + (i + alpha);
	double tmp;
	if (beta <= 2.6e+111) {
		tmp = 0.0625;
	} else if (beta <= 4.75e+142) {
		tmp = ((i * t_1) / fma(t_0, t_0, -1.0)) * ((fma(i, t_1, (beta * alpha)) / t_0) / t_0);
	} else if (beta <= 1.05e+200) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(beta + Float64(i + alpha))
	tmp = 0.0
	if (beta <= 2.6e+111)
		tmp = 0.0625;
	elseif (beta <= 4.75e+142)
		tmp = Float64(Float64(Float64(i * t_1) / fma(t_0, t_0, -1.0)) * Float64(Float64(fma(i, t_1, Float64(beta * alpha)) / t_0) / t_0));
	elseif (beta <= 1.05e+200)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) / Float64(beta + alpha));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6e+111], 0.0625, If[LessEqual[beta, 4.75e+142], N[(N[(N[(i * t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$1 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.05e+200], 0.0625, N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5999999999999999e111 or 4.75e142 < beta < 1.04999999999999999e200

   1. Initial program 19.7%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in i around inf 83.3%

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

   if 2.5999999999999999e111 < beta < 4.75e142

   1. Initial program 21.0%

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

      1. associate-/l/ 1.4%

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

      2. times-frac 59.0%

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

   3. Simplified 59.2%

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

   if 1.04999999999999999e200 < beta

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf 29.1%

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

   6. Taylor expanded in i around 0 47.3%

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

      1. pow1 47.3%

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

      2. un-div-inv 47.4%

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

      3. +-commutative 47.4%

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

      4. +-commutative 47.4%

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

   8. Applied egg-rr 47.4%

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

      1. unpow1 47.4%

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

      2. associate-*r/ 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

   10. Simplified 78.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.75 \cdot 10^{+142}:\\ \;\;\;\;\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.65 \cdot 10^{+142}:\\ \;\;\;\;i \cdot \left(\frac{\mathsf{fma}\left(i, t\_0, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t\_1, t\_1, -1\right)} \cdot \frac{t\_0}{t\_1 \cdot t\_1}\right)\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ beta alpha))) (t_1 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 7e+111)
     0.0625
     (if (<= beta 3.65e+142)
       (*
        i
        (*
         (/ (fma i t_0 (* beta alpha)) (fma t_1 t_1 -1.0))
         (/ t_0 (* t_1 t_1))))
       (if (<= beta 1.05e+200)
         0.0625
         (/ (* i (/ (+ i alpha) beta)) (+ beta alpha)))))))

C

assert(alpha < beta && beta < i);
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 <= 7e+111) {
		tmp = 0.0625;
	} else if (beta <= 3.65e+142) {
		tmp = i * ((fma(i, t_0, (beta * alpha)) / fma(t_1, t_1, -1.0)) * (t_0 / (t_1 * t_1)));
	} else if (beta <= 1.05e+200) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
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 <= 7e+111)
		tmp = 0.0625;
	elseif (beta <= 3.65e+142)
		tmp = Float64(i * Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / fma(t_1, t_1, -1.0)) * Float64(t_0 / Float64(t_1 * t_1))));
	elseif (beta <= 1.05e+200)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) / Float64(beta + alpha));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7e+111], 0.0625, If[LessEqual[beta, 3.65e+142], N[(i * N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.05e+200], 0.0625, N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 7.0000000000000004e111 or 3.64999999999999994e142 < beta < 1.04999999999999999e200

   1. Initial program 19.7%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in i around inf 83.3%

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

   if 7.0000000000000004e111 < beta < 3.64999999999999994e142

   1. Initial program 21.0%

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

   3. Simplified 58.7%

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

   if 1.04999999999999999e200 < beta

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf 29.1%

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

   6. Taylor expanded in i around 0 47.3%

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

      1. pow1 47.3%

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

      2. un-div-inv 47.4%

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

      3. +-commutative 47.4%

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

      4. +-commutative 47.4%

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

   8. Applied egg-rr 47.4%

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

      1. unpow1 47.4%

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

      2. associate-*r/ 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

   10. Simplified 78.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.65 \cdot 10^{+142}:\\ \;\;\;\;i \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+160} \lor \neg \left(\beta \leq 2.55 \cdot 10^{+200}\right):\\ \;\;\;\;i \cdot \left(\frac{i + \alpha}{\beta} \cdot \frac{1}{\beta}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.9e+111)
   0.0625
   (if (or (<= beta 2.6e+160) (not (<= beta 2.55e+200)))
     (* i (* (/ (+ i alpha) beta) (/ 1.0 beta)))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.9e+111) {
		tmp = 0.0625;
	} else if ((beta <= 2.6e+160) || !(beta <= 2.55e+200)) {
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.9d+111) then
        tmp = 0.0625d0
    else if ((beta <= 2.6d+160) .or. (.not. (beta <= 2.55d+200))) then
        tmp = i * (((i + alpha) / beta) * (1.0d0 / beta))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.9e+111) {
		tmp = 0.0625;
	} else if ((beta <= 2.6e+160) || !(beta <= 2.55e+200)) {
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.9e+111:
		tmp = 0.0625
	elif (beta <= 2.6e+160) or not (beta <= 2.55e+200):
		tmp = i * (((i + alpha) / beta) * (1.0 / beta))
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.9e+111)
		tmp = 0.0625;
	elseif ((beta <= 2.6e+160) || !(beta <= 2.55e+200))
		tmp = Float64(i * Float64(Float64(Float64(i + alpha) / beta) * Float64(1.0 / beta)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.9e+111)
		tmp = 0.0625;
	elseif ((beta <= 2.6e+160) || ~((beta <= 2.55e+200)))
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.9e+111], 0.0625, If[Or[LessEqual[beta, 2.6e+160], N[Not[LessEqual[beta, 2.55e+200]], $MachinePrecision]], N[(i * N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999979e111 or 2.6e160 < beta < 2.5499999999999999e200

   1. Initial program 19.7%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in i around inf 83.7%

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

   if 3.89999999999999979e111 < beta < 2.6e160 or 2.5499999999999999e200 < beta

   1. Initial program 3.5%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in beta around inf 33.7%

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

   6. Taylor expanded in beta around inf 48.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+160} \lor \neg \left(\beta \leq 2.55 \cdot 10^{+200}\right):\\ \;\;\;\;i \cdot \left(\frac{i + \alpha}{\beta} \cdot \frac{1}{\beta}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+161} \lor \neg \left(\beta \leq 7.2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.4e+111)
   0.0625
   (if (or (<= beta 2.5e+161) (not (<= beta 7.2e+200)))
     (/ (* i (/ (+ i alpha) beta)) (+ beta alpha))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.4e+111) {
		tmp = 0.0625;
	} else if ((beta <= 2.5e+161) || !(beta <= 7.2e+200)) {
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.4d+111) then
        tmp = 0.0625d0
    else if ((beta <= 2.5d+161) .or. (.not. (beta <= 7.2d+200))) then
        tmp = (i * ((i + alpha) / beta)) / (beta + alpha)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.4e+111) {
		tmp = 0.0625;
	} else if ((beta <= 2.5e+161) || !(beta <= 7.2e+200)) {
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.4e+111:
		tmp = 0.0625
	elif (beta <= 2.5e+161) or not (beta <= 7.2e+200):
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.4e+111)
		tmp = 0.0625;
	elseif ((beta <= 2.5e+161) || !(beta <= 7.2e+200))
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) / Float64(beta + alpha));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.4e+111)
		tmp = 0.0625;
	elseif ((beta <= 2.5e+161) || ~((beta <= 7.2e+200)))
		tmp = (i * ((i + alpha) / beta)) / (beta + alpha);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.4e+111], 0.0625, If[Or[LessEqual[beta, 2.5e+161], N[Not[LessEqual[beta, 7.2e+200]], $MachinePrecision]], N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + alpha), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.3999999999999998e111 or 2.4999999999999998e161 < beta < 7.1999999999999995e200

   1. Initial program 19.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in i around inf 84.1%

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

   if 5.3999999999999998e111 < beta < 2.4999999999999998e161 or 7.1999999999999995e200 < beta

   1. Initial program 3.4%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in beta around inf 32.9%

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

   6. Taylor expanded in i around 0 46.9%

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

      1. pow1 46.9%

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

      2. un-div-inv 47.0%

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

      3. +-commutative 47.0%

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

      4. +-commutative 47.0%

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

   8. Applied egg-rr 47.0%

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

      1. unpow1 47.0%

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

      2. associate-*r/ 71.0%

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

      3. +-commutative 71.0%

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

      4. +-commutative 71.0%

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

   10. Simplified 71.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+161} \lor \neg \left(\beta \leq 7.2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \alpha}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+200) 0.0625 (* i (/ (/ i beta) (+ beta alpha)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+200) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / beta) / (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6d+200) then
        tmp = 0.0625d0
    else
        tmp = i * ((i / beta) / (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+200) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / beta) / (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+200:
		tmp = 0.0625
	else:
		tmp = i * ((i / beta) / (beta + alpha))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+200)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(i / beta) / Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+200)
		tmp = 0.0625;
	else
		tmp = i * ((i / beta) / (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+200], 0.0625, N[(i * N[(N[(i / beta), $MachinePrecision] / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.99999999999999982e200

   1. Initial program 19.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in i around inf 82.5%

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

   if 5.99999999999999982e200 < beta

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf 29.1%

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

   6. Taylor expanded in i around 0 47.3%

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

   7. Taylor expanded in i around inf 29.1%

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

      1. associate-/r* 39.8%

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

   9. Simplified 39.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

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

3. Simplified 40.5%

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

5. Taylor expanded in i around inf 75.8%

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

6. Final simplification 75.8%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(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)))