Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 97.9%

Time: 24.7s

Alternatives: 12

Speedup: 115.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \frac{\left(i + \beta\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right) - 1\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.2e+124)
   (/
    (*
     (* (/ i (fma 2.0 i (+ beta alpha))) (+ i (+ beta alpha)))
     (/
      (* (+ i beta) (/ i (fma 2.0 i beta)))
      (fma 2.0 i (- (+ beta alpha) 1.0))))
    (fma 2.0 i (+ (+ beta alpha) 1.0)))
   (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.2e+124) {
		tmp = (((i / fma(2.0, i, (beta + alpha))) * (i + (beta + alpha))) * (((i + beta) * (i / fma(2.0, i, beta))) / fma(2.0, i, ((beta + alpha) - 1.0)))) / fma(2.0, i, ((beta + alpha) + 1.0));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 8.2e+124)
		tmp = Float64(Float64(Float64(Float64(i / fma(2.0, i, Float64(beta + alpha))) * Float64(i + Float64(beta + alpha))) * Float64(Float64(Float64(i + beta) * Float64(i / fma(2.0, i, beta))) / fma(2.0, i, Float64(Float64(beta + alpha) - 1.0)))) / fma(2.0, i, Float64(Float64(beta + alpha) + 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[alpha, 8.2e+124], N[(N[(N[(N[(i / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i + beta), $MachinePrecision] * N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(beta + alpha), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.20000000000000002e124

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 45.1

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

   7. Applied rewrites 45.1%

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

   9. Applied rewrites 98.2%

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

   11. Applied rewrites 98.2%

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

   if 8.20000000000000002e124 < alpha

   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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 10.2

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

   5. Applied rewrites 10.2%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0} \cdot \frac{i \cdot \frac{i + \beta}{\mathsf{fma}\left(2, i, \beta\right)}}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= alpha 8.2e+124)
     (*
      (/ (* (+ (+ alpha beta) i) (/ i t_0)) (+ 1.0 t_0))
      (/ (* i (/ (+ i beta) (fma 2.0 i beta))) (- t_0 1.0)))
     (* (/ (+ alpha i) beta) (/ i beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (alpha <= 8.2e+124) {
		tmp = ((((alpha + beta) + i) * (i / t_0)) / (1.0 + t_0)) * ((i * ((i + beta) / fma(2.0, i, beta))) / (t_0 - 1.0));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 8.2e+124)
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0)) / Float64(1.0 + t_0)) * Float64(Float64(i * Float64(Float64(i + beta) / fma(2.0, i, beta))) / Float64(t_0 - 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	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[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 8.2e+124], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(N[(i + beta), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.20000000000000002e124

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 45.1

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

   7. Applied rewrites 45.1%

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

   9. Applied rewrites 98.2%

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

   11. Applied rewrites 98.2%

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

   if 8.20000000000000002e124 < alpha

   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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 10.2

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

   5. Applied rewrites 10.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\beta + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= alpha 8.2e+124)
     (*
      (/ (* (+ beta i) (/ i t_0)) (+ 1.0 t_0))
      (/ (* (/ i (fma 2.0 i beta)) (+ beta i)) (- t_0 1.0)))
     (* (/ (+ alpha i) beta) (/ i beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (alpha <= 8.2e+124) {
		tmp = (((beta + i) * (i / t_0)) / (1.0 + t_0)) * (((i / fma(2.0, i, beta)) * (beta + i)) / (t_0 - 1.0));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 8.2e+124)
		tmp = Float64(Float64(Float64(Float64(beta + i) * Float64(i / t_0)) / Float64(1.0 + t_0)) * Float64(Float64(Float64(i / fma(2.0, i, beta)) * Float64(beta + i)) / Float64(t_0 - 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	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[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 8.2e+124], N[(N[(N[(N[(beta + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.20000000000000002e124

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 45.1

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

   7. Applied rewrites 45.1%

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

   9. Applied rewrites 98.2%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 98.1

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

   12. Applied rewrites 98.1%

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

   if 8.20000000000000002e124 < alpha

   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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 10.2

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

   5. Applied rewrites 10.2%

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

Alternative 4: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := 1 + t\_0\\ t_2 := t\_0 - 1\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \left(0.5 + \frac{\left(\alpha + \beta\right) \cdot 0.25}{i}\right)}{t\_1} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}}{t\_1} \cdot \frac{\alpha + i}{t\_2}\\ \end{array} \end{array} \]

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 2.0 i (+ alpha beta))) (t_1 (+ 1.0 t_0)) (t_2 (- t_0 1.0)))
   (if (<= beta 9.8e+134)
     (*
      (/ (* i (+ 0.5 (/ (* (+ alpha beta) 0.25) i))) t_1)
      (/ (* (/ i (fma 2.0 i beta)) (+ beta i)) t_2))
     (* (/ (* (+ (+ alpha beta) i) (/ i t_0)) t_1) (/ (+ alpha i) t_2)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = 1.0 + t_0;
	double t_2 = t_0 - 1.0;
	double tmp;
	if (beta <= 9.8e+134) {
		tmp = ((i * (0.5 + (((alpha + beta) * 0.25) / i))) / t_1) * (((i / fma(2.0, i, beta)) * (beta + i)) / t_2);
	} else {
		tmp = ((((alpha + beta) + i) * (i / t_0)) / t_1) * ((alpha + i) / t_2);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(t_0 - 1.0)
	tmp = 0.0
	if (beta <= 9.8e+134)
		tmp = Float64(Float64(Float64(i * Float64(0.5 + Float64(Float64(Float64(alpha + beta) * 0.25) / i))) / t_1) * Float64(Float64(Float64(i / fma(2.0, i, beta)) * Float64(beta + i)) / t_2));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0)) / t_1) * Float64(Float64(alpha + i) / t_2));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - 1.0), $MachinePrecision]}, If[LessEqual[beta, 9.8e+134], N[(N[(N[(i * N[(0.5 + N[(N[(N[(alpha + beta), $MachinePrecision] * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(beta + i), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.79999999999999992e134

   1. Initial program 18.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 42.6

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

   7. Applied rewrites 42.6%

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

   9. Applied rewrites 88.8%

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

   10. Taylor expanded in i around inf

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

       1. lower-*.f64 N/A

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

       2. associate--l+ N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r/ N/A

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

       5. div-sub N/A

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

       6. lower-+.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. distribute-rgt-out-- N/A

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

       9. metadata-eval N/A

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

       10. lower-*.f64 N/A

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

       11. lower-+.f64 81.3

           \[\leadsto \frac{i \cdot \left(0.5 + \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot 0.25}{i}\right)}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   12. Applied rewrites 81.3%

       \[\leadsto \frac{\color{blue}{i \cdot \left(0.5 + \frac{\left(\alpha + \beta\right) \cdot 0.25}{i}\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   if 9.79999999999999992e134 < beta

   1. Initial program 0.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 12.2%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 14.8

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

   7. Applied rewrites 14.8%

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

   9. Applied rewrites 80.2%

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

   10. Taylor expanded in beta around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 72.5

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

   12. Applied rewrites 72.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \left(0.5 + \frac{\left(\alpha + \beta\right) \cdot 0.25}{i}\right)}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0} \cdot \frac{\alpha + i}{t\_0 - 1}\\ \end{array} \end{array} \]

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 2.0 i (+ alpha beta))))
   (if (<= beta 9.8e+134)
     0.0625
     (*
      (/ (* (+ (+ alpha beta) i) (/ i t_0)) (+ 1.0 t_0))
      (/ (+ alpha i) (- t_0 1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 9.8e+134) {
		tmp = 0.0625;
	} else {
		tmp = ((((alpha + beta) + i) * (i / t_0)) / (1.0 + t_0)) * ((alpha + i) / (t_0 - 1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 9.8e+134)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0)) / Float64(1.0 + t_0)) * Float64(Float64(alpha + 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[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9.8e+134], 0.0625, N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.79999999999999992e134

   1. Initial program 18.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.8%

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

   if 9.79999999999999992e134 < beta

   1. Initial program 0.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

   4. Applied rewrites 12.2%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 14.8

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

   7. Applied rewrites 14.8%

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

   9. Applied rewrites 80.2%

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

   10. Taylor expanded in beta around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 72.5

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

   12. Applied rewrites 72.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\alpha + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.8e+134)
   0.0625
   (/ (* i (/ (+ alpha i) beta)) (- (fma 2.0 i (+ alpha beta)) 1.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.8e+134) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((alpha + i) / beta)) / (fma(2.0, i, (alpha + beta)) - 1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.8e+134)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / beta)) / Float64(fma(2.0, i, Float64(alpha + beta)) - 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_] := If[LessEqual[beta, 9.8e+134], 0.0625, N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.79999999999999992e134

   1. Initial program 18.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.8%

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

   if 9.79999999999999992e134 < beta

   1. Initial program 0.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 7.0%

      \[\leadsto \frac{\color{blue}{\left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\right) \cdot \left(i \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 7.0%

      \[\leadsto \frac{\left(0.25 \cdot \left(i + \left(\left(2 \cdot \left(\beta + \alpha\right) - \alpha\right) - \beta\right)\right)\right) \cdot \color{blue}{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   10. Applied rewrites 3.6%

       \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(\left(\left(\alpha + \beta\right) \cdot 2 - \alpha\right) - \beta\right) + i\right) \cdot 0.25\right) \cdot i}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}} \]

   11. Taylor expanded in beta around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. associate-/l* N/A

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

       4. distribute-lft-out N/A

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

       5. associate-*r/ N/A

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

       6. lower-*.f64 N/A

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

       7. div-add-rev N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. div-add-rev N/A

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

       11. lower-/.f64 N/A

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

       12. lower-+.f64 70.4

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

   13. Applied rewrites 70.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.8e+134) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.8e+134) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 9.8d+134) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.8e+134) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.8e+134:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.8e+134)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.8e+134)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 9.8e+134], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.79999999999999992e134

   1. Initial program 18.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.8%

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

   if 9.79999999999999992e134 < beta

   1. Initial program 0.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

Alternative 8: 82.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.24 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.24e+135) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.24e+135) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.24d+135) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.24e+135) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.24e+135:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.24e+135)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.24e+135)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.24e+135], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.23999999999999993e135

   1. Initial program 18.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.8%

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

   if 1.23999999999999993e135 < beta

   1. Initial program 0.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 64.4%

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

Alternative 9: 74.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+196}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.5e+196) 0.0625 (* (/ alpha beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+196) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 8.5d+196) then
        tmp = 0.0625d0
    else
        tmp = (alpha / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+196) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+196:
		tmp = 0.0625
	else:
		tmp = (alpha / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+196)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+196)
		tmp = 0.0625;
	else
		tmp = (alpha / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+196], 0.0625, N[(N[(alpha / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.50000000000000041e196

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 76.2%

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

   if 8.50000000000000041e196 < 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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 48.0%

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

Alternative 10: 74.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+214}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.5e+214) 0.0625 (* alpha (/ (/ i beta) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.5e+214) {
		tmp = 0.0625;
	} else {
		tmp = alpha * ((i / beta) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.5d+214) then
        tmp = 0.0625d0
    else
        tmp = alpha * ((i / beta) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.5e+214) {
		tmp = 0.0625;
	} else {
		tmp = alpha * ((i / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.5e+214:
		tmp = 0.0625
	else:
		tmp = alpha * ((i / beta) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.5e+214)
		tmp = 0.0625;
	else
		tmp = Float64(alpha * Float64(Float64(i / beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.5e+214)
		tmp = 0.0625;
	else
		tmp = alpha * ((i / beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.5e+214], 0.0625, N[(alpha * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e214

   1. Initial program 16.4%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.3%

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

   if 3.5e214 < 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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 43.2%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 43.2%

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

   12. Applied rewrites 45.3%

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

Alternative 11: 74.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.9 \cdot 10^{+234}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.9e+234) 0.0625 (* alpha (/ i (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.9e+234) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * 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 <= 4.9d+234) then
        tmp = 0.0625d0
    else
        tmp = alpha * (i / (beta * 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 <= 4.9e+234) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.9e+234:
		tmp = 0.0625
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.9e+234)
		tmp = 0.0625;
	else
		tmp = Float64(alpha * Float64(i / Float64(beta * 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 <= 4.9e+234)
		tmp = 0.0625;
	else
		tmp = alpha * (i / (beta * 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, 4.9e+234], 0.0625, N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.89999999999999989e234

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 74.7%

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

   if 4.89999999999999989e234 < 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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   7. Applied rewrites 46.2%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 46.2%

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

Alternative 12: 71.1% accurate, 115.0× 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 14.7%

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

3. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
4. Step-by-step derivation

5. Applied rewrites 68.6%

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

Reproduce

herbie shell --seed 2024331 
(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)))