Octave 3.8, jcobi/4

Percentage Accurate: 15.6% → 85.5%

Time: 14.5s

Alternatives: 9

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: 15.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)} \cdot \frac{i + \alpha}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+160)
   0.0625
   (*
    (/ (* i (- 1.0 (/ i (+ beta alpha)))) (+ (fma i 2.0 beta) (+ alpha 1.0)))
    (/ (+ i alpha) (+ alpha (+ (fma i 2.0 beta) -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+160) {
		tmp = 0.0625;
	} else {
		tmp = ((i * (1.0 - (i / (beta + alpha)))) / (fma(i, 2.0, beta) + (alpha + 1.0))) * ((i + alpha) / (alpha + (fma(i, 2.0, beta) + -1.0)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+160)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i * Float64(1.0 - Float64(i / Float64(beta + alpha)))) / Float64(fma(i, 2.0, beta) + Float64(alpha + 1.0))) * Float64(Float64(i + alpha) / Float64(alpha + Float64(fma(i, 2.0, 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, 2e+160], 0.0625, N[(N[(N[(i * N[(1.0 - N[(i / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000001e160

   1. Initial program 19.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.9%

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

   if 2.00000000000000001e160 < 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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 20.3

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

   7. Applied rewrites 20.3%

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

   8. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 20.3

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

   10. Applied rewrites 20.3%

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

   12. Applied rewrites 76.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)} \cdot \frac{i + \alpha}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_3 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \beta \cdot \alpha\right)}{t\_1}}{-1 + t\_1} \leq 10^{-15}:\\ \;\;\;\;\frac{\beta \cdot i}{t\_3} \cdot \frac{i + \alpha}{\mathsf{fma}\left(t\_3, t\_3, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha))))
        (t_3 (fma i 2.0 (+ beta alpha))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1)) 1e-15)
     (* (/ (* beta i) t_3) (/ (+ i alpha) (fma t_3 t_3 -1.0)))
     (+
      (fma 0.0625 (/ (* 2.0 (+ beta alpha)) i) 0.0625)
      (* -0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double t_3 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if ((((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1)) <= 1e-15) {
		tmp = ((beta * i) / t_3) * ((i + alpha) / fma(t_3, t_3, -1.0));
	} else {
		tmp = fma(0.0625, ((2.0 * (beta + alpha)) / i), 0.0625) + (-0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	t_3 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1)) <= 1e-15)
		tmp = Float64(Float64(Float64(beta * i) / t_3) * Float64(Float64(i + alpha) / fma(t_3, t_3, -1.0)));
	else
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-15], N[(N[(N[(beta * i), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in beta around inf

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

      1. lower-*.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if 1.0000000000000001e-15 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.8%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i}}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \frac{-1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      12. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \leq 10^{-15}:\\ \;\;\;\;\frac{\beta \cdot i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \beta \cdot \alpha\right)}{t\_1}}{-1 + t\_1} \leq 10^{-15}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(4, i \cdot \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, \beta + \alpha, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1)) 1e-15)
     (*
      (* i (- 1.0 (/ i (+ beta alpha))))
      (/
       (+ i alpha)
       (fma
        4.0
        (* i (+ beta alpha))
        (fma (+ beta alpha) (+ beta alpha) -1.0))))
     (+
      (fma 0.0625 (/ (* 2.0 (+ beta alpha)) i) 0.0625)
      (* -0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double tmp;
	if ((((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1)) <= 1e-15) {
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / fma(4.0, (i * (beta + alpha)), fma((beta + alpha), (beta + alpha), -1.0)));
	} else {
		tmp = fma(0.0625, ((2.0 * (beta + alpha)) / i), 0.0625) + (-0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1)) <= 1e-15)
		tmp = Float64(Float64(i * Float64(1.0 - Float64(i / Float64(beta + alpha)))) * Float64(Float64(i + alpha) / fma(4.0, Float64(i * Float64(beta + alpha)), fma(Float64(beta + alpha), Float64(beta + alpha), -1.0))));
	else
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-15], N[(N[(i * N[(1.0 - N[(i / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(4.0 * N[(i * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 53.9

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

   10. Applied rewrites 53.9%

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

   11. Taylor expanded in i around 0

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

       1. associate--l+ N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 53.9

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

   13. Applied rewrites 53.9%

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

   if 1.0000000000000001e-15 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.8%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i}}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \frac{-1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      12. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \leq 10^{-15}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(4, i \cdot \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, \beta + \alpha, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \beta \cdot \alpha\right)}{t\_1}}{-1 + t\_1} \leq 10^{-15}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta\right), \mathsf{fma}\left(2, i, \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1)) 1e-15)
     (*
      (* i (- 1.0 (/ i (+ beta alpha))))
      (/ (+ i alpha) (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0)))
     (+
      (fma 0.0625 (/ (* 2.0 (+ beta alpha)) i) 0.0625)
      (* -0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double tmp;
	if ((((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1)) <= 1e-15) {
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0));
	} else {
		tmp = fma(0.0625, ((2.0 * (beta + alpha)) / i), 0.0625) + (-0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1)) <= 1e-15)
		tmp = Float64(Float64(i * Float64(1.0 - Float64(i / Float64(beta + alpha)))) * Float64(Float64(i + alpha) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)));
	else
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-15], N[(N[(i * N[(1.0 - N[(i / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 53.9

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

   10. Applied rewrites 53.9%

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

   11. Taylor expanded in alpha around 0

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

       1. sub-neg N/A

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

       2. unpow2 N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 52.8

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

   13. Applied rewrites 52.8%

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

   if 1.0000000000000001e-15 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.8%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i}}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \frac{-1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      12. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \leq 10^{-15}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta\right), \mathsf{fma}\left(2, i, \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := -1 + t\_1\\ t_3 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \beta \cdot \alpha\right)}{t\_1}}{t\_2} \leq 10^{-15}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ -1.0 t_1))
        (t_3 (* i (+ i (+ beta alpha)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* beta alpha))) t_1) t_2) 1e-15)
     (/ (* i (+ i alpha)) t_2)
     (+
      (fma 0.0625 (/ (* 2.0 (+ beta alpha)) i) 0.0625)
      (* -0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = -1.0 + t_1;
	double t_3 = i * (i + (beta + alpha));
	double tmp;
	if ((((t_3 * (t_3 + (beta * alpha))) / t_1) / t_2) <= 1e-15) {
		tmp = (i * (i + alpha)) / t_2;
	} else {
		tmp = fma(0.0625, ((2.0 * (beta + alpha)) / i), 0.0625) + (-0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(-1.0 + t_1)
	t_3 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(beta * alpha))) / t_1) / t_2) <= 1e-15)
		tmp = Float64(Float64(i * Float64(i + alpha)) / t_2);
	else
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 1e-15], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.0000000000000001e-15

   1. Initial program 98.3%

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

   3. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 53.9

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

   5. Applied rewrites 53.9%

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

   if 1.0000000000000001e-15 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.8%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i}}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \frac{-1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      12. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \leq 10^{-15}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \beta \cdot \alpha\right)}{t\_1}}{-1 + t\_1} \leq 10^{-15}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ -1.0 t_1)) 1e-15)
     (/ (* i (+ i alpha)) (* beta beta))
     (+
      (fma 0.0625 (/ (* 2.0 (+ beta alpha)) i) 0.0625)
      (* -0.125 (/ (+ beta alpha) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double tmp;
	if ((((t_2 * (t_2 + (beta * alpha))) / t_1) / (-1.0 + t_1)) <= 1e-15) {
		tmp = (i * (i + alpha)) / (beta * beta);
	} else {
		tmp = fma(0.0625, ((2.0 * (beta + alpha)) / i), 0.0625) + (-0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(-1.0 + t_1)) <= 1e-15)
		tmp = Float64(Float64(i * Float64(i + alpha)) / Float64(beta * beta));
	else
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625) + Float64(-0.125 * Float64(Float64(beta + alpha) / i)));
	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[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-15], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 1.0000000000000001e-15

   1. Initial program 98.3%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   if 1.0000000000000001e-15 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.8%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \alpha + 2 \cdot \beta}{i}, \frac{1}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\frac{2 \cdot \alpha + 2 \cdot \beta}{i}}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \color{blue}{\left(\alpha + \beta\right)}}{i}, \frac{1}{16}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\alpha + \beta}{i} \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \color{blue}{\frac{-1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16}, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, \frac{1}{16}\right) + \frac{-1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      12. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \leq 10^{-15}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+211}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+211)
   0.0625
   (* (* i (- 1.0 (/ i (+ beta alpha)))) (/ (+ i alpha) (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+211) {
		tmp = 0.0625;
	} else {
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / (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 <= 1d+211) then
        tmp = 0.0625d0
    else
        tmp = (i * (1.0d0 - (i / (beta + alpha)))) * ((i + alpha) / (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 <= 1e+211) {
		tmp = 0.0625;
	} else {
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1e+211:
		tmp = 0.0625
	else:
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+211)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(1.0 - Float64(i / Float64(beta + alpha)))) * Float64(Float64(i + alpha) / 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 <= 1e+211)
		tmp = 0.0625;
	else
		tmp = (i * (1.0 - (i / (beta + alpha)))) * ((i + alpha) / (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, 1e+211], 0.0625, N[(N[(i * N[(1.0 - N[(i / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.9999999999999996e210

   1. Initial program 18.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 74.8%

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

   if 9.9999999999999996e210 < 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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 16.7

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 39.6

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

   10. Applied rewrites 39.6%

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

   11. Taylor expanded in beta around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 39.6

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

   13. Applied rewrites 39.6%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+211}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(1 - \frac{i}{\beta + \alpha}\right)\right) \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+233) 0.0625 (/ (* i (+ i alpha)) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+233) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i + alpha)) / (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 <= 1d+233) then
        tmp = 0.0625d0
    else
        tmp = (i * (i + alpha)) / (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 <= 1e+233) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i + alpha)) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1e+233:
		tmp = 0.0625
	else:
		tmp = (i * (i + alpha)) / (beta * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+233)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i + alpha)) / 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 <= 1e+233)
		tmp = 0.0625;
	else
		tmp = (i * (i + alpha)) / (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, 1e+233], 0.0625, N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999974e232

   1. Initial program 18.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.2%

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

   if 9.99999999999999974e232 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 16.5%

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

3. 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 2024212 
(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)))