Octave 3.8, jcobi/4

Percentage Accurate: 16.9% → 97.0%

Time: 13.0s

Alternatives: 12

Speedup: 115.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \frac{\left(i + \beta\right) \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta\right) - 1} \cdot \frac{\frac{i}{t\_0} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	return Float64(Float64(Float64(Float64(i + beta) * Float64(i / fma(i, 2.0, beta))) / Float64(fma(i, 2.0, beta) - 1.0)) * Float64(Float64(Float64(i / t_0) * Float64(i + Float64(alpha + beta))) / Float64(1.0 + t_0)))
end

Wolfram

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

Derivation

1. Initial program 19.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.2

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 19.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 19.2

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

4. Applied rewrites 19.2%

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

6. Applied rewrites 46.3%

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

7. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 39.1

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

9. Applied rewrites 39.1%

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

11. Applied rewrites 85.9%

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

12. Final simplification 85.9%

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

Alternative 2: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) - 1} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \frac{\frac{i}{t\_0} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	return Float64(Float64(Float64(i / Float64(fma(i, 2.0, beta) - 1.0)) * Float64(Float64(i + beta) / fma(i, 2.0, beta))) * Float64(Float64(Float64(i / t_0) * Float64(i + Float64(alpha + beta))) / Float64(1.0 + t_0)))
end

Wolfram

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

Derivation

1. Initial program 19.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.2

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 19.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 19.2

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

4. Applied rewrites 19.2%

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

6. Applied rewrites 46.3%

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

7. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 39.1

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

9. Applied rewrites 39.1%

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

11. Applied rewrites 85.9%

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

12. Final simplification 85.9%

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

Alternative 3: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_0 - 1} \cdot \frac{\frac{i}{t\_0} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= beta 3.6e+216)
     (-
      (fma 0.0625 (/ (* 2.0 (+ alpha beta)) i) 0.0625)
      (* (/ (+ alpha beta) i) 0.125))
     (*
      (/ (+ i alpha) (- t_0 1.0))
      (/ (* (/ i t_0) (+ i (+ alpha beta))) (+ 1.0 t_0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = fma(0.0625, ((2.0 * (alpha + beta)) / i), 0.0625) - (((alpha + beta) / i) * 0.125);
	} else {
		tmp = ((i + alpha) / (t_0 - 1.0)) * (((i / t_0) * (i + (alpha + beta))) / (1.0 + t_0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(alpha + beta)) / i), 0.0625) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(t_0 - 1.0)) * Float64(Float64(Float64(i / t_0) * Float64(i + Float64(alpha + beta))) / Float64(1.0 + t_0)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.6e+216], N[(N[(0.0625 * N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 21.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 21.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 21.1

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

   4. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 21.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 21.2

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

   6. Applied rewrites 21.2%

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

   7. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if 3.6000000000000002e216 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 0.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 0.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 10.7%

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

   7. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 83.5

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

   9. Applied rewrites 83.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{\frac{i}{t\_0} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= beta 3.6e+216)
     (-
      (fma 0.0625 (/ (* 2.0 (+ alpha beta)) i) 0.0625)
      (* (/ (+ alpha beta) i) 0.125))
     (*
      (/ (+ i alpha) beta)
      (/ (* (/ i t_0) (+ i (+ alpha beta))) (+ 1.0 t_0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = fma(0.0625, ((2.0 * (alpha + beta)) / i), 0.0625) - (((alpha + beta) / i) * 0.125);
	} else {
		tmp = ((i + alpha) / beta) * (((i / t_0) * (i + (alpha + beta))) / (1.0 + t_0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(alpha + beta)) / i), 0.0625) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(Float64(Float64(i / t_0) * Float64(i + Float64(alpha + beta))) / Float64(1.0 + t_0)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.6e+216], N[(N[(0.0625 * N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 21.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 21.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 21.1

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

   4. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 21.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 21.2

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

   6. Applied rewrites 21.2%

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

   7. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if 3.6000000000000002e216 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 0.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 0.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 10.7%

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

   7. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 82.5

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

   9. Applied rewrites 82.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216)
   (-
    (fma 0.0625 (/ (* 2.0 (+ alpha beta)) i) 0.0625)
    (* (/ (+ alpha beta) i) 0.125))
   (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = fma(0.0625, ((2.0 * (alpha + beta)) / i), 0.0625) - (((alpha + beta) / i) * 0.125);
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = Float64(fma(0.0625, Float64(Float64(2.0 * Float64(alpha + beta)) / i), 0.0625) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / 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_] := If[LessEqual[beta, 3.6e+216], N[(N[(0.0625 * N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 21.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 21.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 21.1

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

   4. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 21.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 21.2

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

   6. Applied rewrites 21.2%

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

   7. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{2 \cdot \left(\alpha + \beta\right)}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\beta}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216)
   (- (fma 0.125 (/ beta i) 0.0625) (* (/ (+ alpha beta) i) 0.125))
   (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = fma(0.125, (beta / i), 0.0625) - (((alpha + beta) / i) * 0.125);
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = Float64(fma(0.125, Float64(beta / i), 0.0625) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / 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_] := If[LessEqual[beta, 3.6e+216], N[(N[(0.125 * N[(beta / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 21.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 21.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 21.1

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

   4. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 21.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 21.2

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

   6. Applied rewrites 21.2%

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

   7. Taylor expanded in i around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 82.4

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

   9. Applied rewrites 82.4%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 77.6%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\beta}{i}, 0.0625\right) - \frac{\alpha + \beta}{i} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.6d+216) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.6e+216:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.6e+216], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.7%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.6d+216) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.6e+216:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.6e+216], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.7%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.6d+216) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.6e+216:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.6e+216], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.7%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 78.4%

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

Alternative 10: 76.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+216}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.6e+216) 0.0625 (* (/ (/ i beta) beta) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.6d+216) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) / beta) * i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.6e+216) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.6e+216:
		tmp = 0.0625
	else:
		tmp = ((i / beta) / beta) * i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) / beta) * i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.6e+216)
		tmp = 0.0625;
	else
		tmp = ((i / beta) / beta) * i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.6e+216], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6000000000000002e216

   1. Initial program 21.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 78.7%

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

   if 3.6000000000000002e216 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 27.5%

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

   12. Applied rewrites 41.9%

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

Alternative 11: 73.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+267}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3e+267) 0.0625 (* (/ i (* beta beta)) alpha)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3e+267) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3d+267) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta * beta)) * alpha
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3e+267) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3e+267:
		tmp = 0.0625
	else:
		tmp = (i / (beta * beta)) * alpha
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3e+267)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta * beta)) * alpha);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3e+267)
		tmp = 0.0625;
	else
		tmp = (i / (beta * beta)) * alpha;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3e+267], 0.0625, N[(N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9999999999999999e267

   1. Initial program 20.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 76.2%

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

   if 2.9999999999999999e267 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 42.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+267}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.3% 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 19.3%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 72.5%

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

Reproduce

herbie shell --seed 2024283 
(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)))