Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 85.4%

Time: 20.0s

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: 16.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + i\right)\\ t_1 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_2 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+107}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 10^{+137}:\\ \;\;\;\;\left(\mathsf{fma}\left(\beta, \alpha, i \cdot t\_0\right) \cdot {t\_1}^{-2}\right) \cdot \frac{i}{\frac{{t\_1}^{2} + -1}{t\_0}}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+158}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{t\_2 + 1} \cdot \frac{i}{t\_2 + -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta i)))
        (t_1 (+ alpha (fma i 2.0 beta)))
        (t_2 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 1.85e+107)
     0.0625
     (if (<= beta 1e+137)
       (*
        (* (fma beta alpha (* i t_0)) (pow t_1 -2.0))
        (/ i (/ (+ (pow t_1 2.0) -1.0) t_0)))
       (if (<= beta 6.4e+158)
         (+
          (+ 0.0625 (* 0.0625 (/ (* 2.0 (+ beta alpha)) i)))
          (* -0.125 (/ (+ beta alpha) i)))
         (* (/ (+ alpha i) (+ t_2 1.0)) (/ i (+ t_2 -1.0))))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + (beta + i);
	double t_1 = alpha + fma(i, 2.0, beta);
	double t_2 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 1.85e+107) {
		tmp = 0.0625;
	} else if (beta <= 1e+137) {
		tmp = (fma(beta, alpha, (i * t_0)) * pow(t_1, -2.0)) * (i / ((pow(t_1, 2.0) + -1.0) / t_0));
	} else if (beta <= 6.4e+158) {
		tmp = (0.0625 + (0.0625 * ((2.0 * (beta + alpha)) / i))) + (-0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + i) / (t_2 + 1.0)) * (i / (t_2 + -1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(beta + i))
	t_1 = Float64(alpha + fma(i, 2.0, beta))
	t_2 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.85e+107)
		tmp = 0.0625;
	elseif (beta <= 1e+137)
		tmp = Float64(Float64(fma(beta, alpha, Float64(i * t_0)) * (t_1 ^ -2.0)) * Float64(i / Float64(Float64((t_1 ^ 2.0) + -1.0) / t_0)));
	elseif (beta <= 6.4e+158)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(2.0 * Float64(beta + alpha)) / i))) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(t_2 + 1.0)) * Float64(i / Float64(t_2 + -1.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(beta + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.85e+107], 0.0625, If[LessEqual[beta, 1e+137], N[(N[(N[(beta * alpha + N[(i * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] + -1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.4e+158], N[(N[(0.0625 + N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.85e107

   1. Initial program 21.7%

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

      1. associate-/l/ 19.5%

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

      2. associate-*l* 19.4%

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

      3. times-frac 28.7%

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

   3. Simplified 28.7%

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

   5. Taylor expanded in i around inf 85.5%

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

   if 1.85e107 < beta < 1e137

   1. Initial program 1.1%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 41.2%

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

   3. Simplified 41.2%

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

      1. add-log-exp 16.7%

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

      2. exp-prod 25.1%

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

      3. associate-/l* 28.2%

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

      4. fma-udef 28.2%

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

      5. pow2 28.2%

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

      6. +-commutative 28.2%

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

      7. associate-+l+ 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. log-pow 17.2%

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

      2. fma-def 17.2%

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

      3. +-commutative 17.2%

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

      4. fma-def 17.2%

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

      5. rem-log-exp 41.3%

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

      6. +-commutative 41.3%

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

   8. Simplified 41.3%

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

   if 1e137 < beta < 6.39999999999999989e158

   1. Initial program 0.3%

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

      1. associate-/l/ 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 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in i around inf 59.1%

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

      1. cancel-sign-sub-inv 59.1%

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

      2. distribute-lft-out 59.1%

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

      3. metadata-eval 59.1%

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

   7. Simplified 59.1%

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

   if 6.39999999999999989e158 < 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. Add Preprocessing

   3. Taylor expanded in beta around -inf 23.4%

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

      1. associate-*r* 23.4%

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

      2. mul-1-neg 23.4%

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

      3. distribute-lft-out 23.4%

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

   5. Simplified 23.4%

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

      1. *-commutative 23.4%

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

      2. difference-of-sqr-1 23.4%

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

      3. times-frac 76.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 21.2%

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

      6. mul-1-neg 21.2%

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

      7. mul-1-neg 21.2%

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

      8. sqr-neg 21.2%

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

      9. sqrt-unprod 24.4%

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

      10. add-sqr-sqrt 24.4%

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

      11. +-commutative 24.4%

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

      12. *-commutative 24.4%

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

      13. fma-def 24.4%

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

      14. +-commutative 24.4%

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

   7. Applied egg-rr 76.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+107}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 10^{+137}:\\ \;\;\;\;\left(\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\alpha + \left(\beta + i\right)\right)\right) \cdot {\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{-2}\right) \cdot \frac{i}{\frac{{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}{\alpha + \left(\beta + i\right)}}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+158}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{t\_0 + 1} \cdot \frac{i}{t\_0 + -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 4e+158)
     0.0625
     (* (/ (+ alpha i) (+ t_0 1.0)) (/ i (+ t_0 -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 4e+158) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / (t_0 + 1.0)) * (i / (t_0 + -1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(t_0 + 1.0)) * Float64(i / Float64(t_0 + -1.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+158], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999981e158

   1. Initial program 19.8%

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

      1. associate-/l/ 17.8%

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

      2. associate-*l* 17.7%

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

      3. times-frac 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in i around inf 82.2%

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

   if 3.99999999999999981e158 < 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. Add Preprocessing

   3. Taylor expanded in beta around -inf 23.4%

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

      1. associate-*r* 23.4%

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

      2. mul-1-neg 23.4%

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

      3. distribute-lft-out 23.4%

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

   5. Simplified 23.4%

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

      1. *-commutative 23.4%

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

      2. difference-of-sqr-1 23.4%

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

      3. times-frac 76.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 21.2%

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

      6. mul-1-neg 21.2%

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

      7. mul-1-neg 21.2%

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

      8. sqr-neg 21.2%

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

      9. sqrt-unprod 24.4%

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

      10. add-sqr-sqrt 24.4%

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

      11. +-commutative 24.4%

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

      12. *-commutative 24.4%

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

      13. fma-def 24.4%

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

      14. +-commutative 24.4%

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

   7. Applied egg-rr 76.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \left(\left(\alpha + i\right) \cdot \frac{1}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8e+158) 0.0625 (* (/ i beta) (* (+ alpha i) (/ 1.0 beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) * (1.0 / beta));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 8d+158) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + i) * (1.0d0 / beta))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) * (1.0 / beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8e+158:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) * (1.0 / beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) * Float64(1.0 / beta)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8e+158)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + i) * (1.0 / beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8e+158], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.99999999999999962e158

   1. Initial program 19.8%

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

      1. associate-/l/ 17.8%

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

      2. associate-*l* 17.7%

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

      3. times-frac 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in i around inf 82.2%

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

   if 7.99999999999999962e158 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 10.7%

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

   3. Simplified 10.7%

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

   5. Taylor expanded in beta around inf 17.1%

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

   6. Taylor expanded in beta around inf 74.4%

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

      1. div-inv 74.5%

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

   8. Applied egg-rr 74.5%

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

      1. +-commutative 74.5%

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

   10. Simplified 74.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \left(\left(\alpha + i\right) \cdot \frac{1}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7e+158) 0.0625 (* (/ i beta) (/ (+ alpha i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7d+158) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + i) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7e+158:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7e+158)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + i) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7e+158], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.0000000000000003e158

   1. Initial program 19.8%

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

      1. associate-/l/ 17.8%

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

      2. associate-*l* 17.7%

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

      3. times-frac 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in i around inf 82.2%

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

   if 7.0000000000000003e158 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 10.7%

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

   3. Simplified 10.7%

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

   5. Taylor expanded in beta around inf 17.1%

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

   6. Taylor expanded in beta around inf 74.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.6e+232) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.6e+232) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.6d+232) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.6e+232) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.6e+232:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.6e+232)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.6e+232)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.6e+232], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.6e232

   1. Initial program 18.2%

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

      1. associate-/l/ 16.3%

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

      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 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in i around inf 78.5%

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

   if 6.6e232 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. times-frac 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in beta around inf 17.2%

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

   6. Taylor expanded in beta around inf 82.1%

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

   7. Taylor expanded in alpha around inf 39.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.2%

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

   1. associate-/l/ 14.5%

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

   2. associate-*l* 14.4%

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

   3. times-frac 21.7%

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

3. Simplified 21.7%

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

5. Taylor expanded in i around inf 71.5%

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

6. Final simplification 71.5%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))