Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.5%

Time: 10.9s

Alternatives: 20

Speedup: 2.6×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\frac{{\left(\left(\alpha + \beta\right) + 2\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot {\beta}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+161)
   (/
    (*
     (pow (+ (+ alpha beta) 2.0) -2.0)
     (+ 1.0 (fma beta alpha (+ alpha beta))))
    (+ (+ alpha beta) 3.0))
   (* (/ alpha beta) (pow beta -1.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+161) {
		tmp = (pow(((alpha + beta) + 2.0), -2.0) * (1.0 + fma(beta, alpha, (alpha + beta)))) / ((alpha + beta) + 3.0);
	} else {
		tmp = (alpha / beta) * pow(beta, -1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+161)
		tmp = Float64(Float64((Float64(Float64(alpha + beta) + 2.0) ^ -2.0) * Float64(1.0 + fma(beta, alpha, Float64(alpha + beta)))) / Float64(Float64(alpha + beta) + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) * (beta ^ -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6e+161], N[(N[(N[Power[N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision], -2.0], $MachinePrecision] * N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] * N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000023e161

   1. Initial program 97.6%

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

   4. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}} \]

   if 6.00000000000000023e161 < beta

   1. Initial program 80.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 91.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 91.6%

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

   10. Applied rewrites 90.7%

       \[\leadsto {\beta}^{-1} \cdot \frac{\alpha}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\frac{{\left(\left(\alpha + \beta\right) + 2\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot {\beta}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, \left(\alpha + \beta\right) + 3, \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.4e+49)
   (/
    (/ (+ 1.0 (fma beta alpha (+ alpha beta))) (+ (+ alpha beta) 2.0))
    (fma (+ alpha beta) (+ (+ alpha beta) 3.0) (fma 2.0 (+ alpha beta) 6.0)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.4e+49) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / ((alpha + beta) + 2.0)) / fma((alpha + beta), ((alpha + beta) + 3.0), fma(2.0, (alpha + beta), 6.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.4e+49)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / Float64(Float64(alpha + beta) + 2.0)) / fma(Float64(alpha + beta), Float64(Float64(alpha + beta) + 3.0), fma(2.0, Float64(alpha + beta), 6.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.4e+49], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] + N[(2.0 * N[(alpha + beta), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.3999999999999995e49

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 98.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   if 9.3999999999999995e49 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 85.4

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.4%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, \left(\alpha + \beta\right) + 3, \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{\left(\alpha + \beta\right) + 3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{\alpha - -1} \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 5e+174)
     (/
      (/
       (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
       (+ (+ alpha beta) 3.0))
      t_0)
     (/ 1.0 (* (/ beta (- alpha -1.0)) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5e+174) {
		tmp = (((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / ((alpha + beta) + 3.0)) / t_0;
	} else {
		tmp = 1.0 / ((beta / (alpha - -1.0)) * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 5e+174)
		tmp = Float64(Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(Float64(alpha + beta) + 3.0)) / t_0);
	else
		tmp = Float64(1.0 / Float64(Float64(beta / Float64(alpha - -1.0)) * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+174], N[(N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[(beta / N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999997e174

   1. Initial program 97.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   if 4.9999999999999997e174 < beta

   1. Initial program 81.8%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 96.8

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\beta}{\alpha - -1} \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 9.4e+49)
     (/
      (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
      (* (+ (+ alpha beta) 3.0) t_0))
     (/ (/ (- alpha -1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.4e+49) {
		tmp = ((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / (((alpha + beta) + 3.0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 9.4e+49)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(Float64(Float64(alpha + beta) + 3.0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9.4e+49], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.3999999999999995e49

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   if 9.3999999999999995e49 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 85.4

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.4%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\mathsf{fma}\left(4 + \alpha, \alpha, 4\right)}}{1 + t\_0}\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ (+ alpha beta) 3.0)))
   (if (<= beta 4.8e-10)
     (/ (/ (- alpha -1.0) (fma (+ 4.0 alpha) alpha 4.0)) (+ 1.0 t_0))
     (if (<= beta 1.2e+16)
       (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* t_1 t_0))
       (/ (/ (- alpha -1.0) t_1) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 4.8e-10) {
		tmp = ((alpha - -1.0) / fma((4.0 + alpha), alpha, 4.0)) / (1.0 + t_0);
	} else if (beta <= 1.2e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_1 * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 4.8e-10)
		tmp = Float64(Float64(Float64(alpha - -1.0) / fma(Float64(4.0 + alpha), alpha, 4.0)) / Float64(1.0 + t_0));
	elseif (beta <= 1.2e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 4.8e-10], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(4.0 + alpha), $MachinePrecision] * alpha + 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.2e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.8e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.1%

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

   if 4.8e-10 < beta < 1.2e16

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

      3. lower-+.f64 89.9

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   7. Applied rewrites 89.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   if 1.2e16 < beta

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\mathsf{neg}\left(\left(\alpha + 1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{-\left(\mathsf{neg}\left(\color{blue}{\left(1 + \alpha\right)}\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      7. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      9. unsub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      10. lower--.f64 83.0

          \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 83.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\mathsf{fma}\left(4 + \alpha, \alpha, 4\right)}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\left(\left(\alpha + \beta\right) + 3\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 9.4e+49)
     (/
      (+ 1.0 (fma beta alpha (+ alpha beta)))
      (* (* (+ (+ alpha beta) 3.0) t_0) t_0))
     (/ (/ (- alpha -1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 9.4e+49) {
		tmp = (1.0 + fma(beta, alpha, (alpha + beta))) / ((((alpha + beta) + 3.0) * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 9.4e+49)
		tmp = Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / Float64(Float64(Float64(Float64(alpha + beta) + 3.0) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9.4e+49], N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.3999999999999995e49

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   if 9.3999999999999995e49 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 85.4

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.4%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\left(\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 6.2e-15)
     (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
     (if (<= beta 8.2e+15)
       (/ (/ (+ 1.0 beta) (fma (+ 5.0 beta) beta 6.0)) t_0)
       (/ (/ (- alpha -1.0) (+ (+ alpha beta) 3.0)) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 6.2e-15) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / fma((5.0 + beta), beta, 6.0)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 6.2e-15)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	elseif (beta <= 8.2e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / fma(Float64(5.0 + beta), beta, 6.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 3.0)) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 6.2e-15], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8.2e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.1999999999999998e-15

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 92.3

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 92.3%

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 6.1999999999999998e-15 < beta < 8.2e15

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}{\left(\beta + \alpha\right) + 2} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(3 + \beta\right)} \cdot \left(2 + \beta\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. lower-+.f64 68.0

         \[\leadsto \frac{\frac{1 + \beta}{\left(3 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 68.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1 + \beta}{6 + \color{blue}{\beta \cdot \left(5 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.2%

       \[\leadsto \frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \color{blue}{\beta}, 6\right)}}{\left(\beta + \alpha\right) + 2} \]

   if 8.2e15 < beta

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\mathsf{neg}\left(\left(\alpha + 1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{-\left(\mathsf{neg}\left(\color{blue}{\left(1 + \alpha\right)}\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      7. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      9. unsub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      10. lower--.f64 83.0

          \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 83.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2e-15)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (if (<= beta 5.6e+15)
     (/ (+ 1.0 beta) (* (fma beta (+ (+ 3.0 beta) 2.0) 6.0) (+ 2.0 beta)))
     (/ (/ (- alpha -1.0) (+ (+ alpha beta) 3.0)) (+ (+ alpha beta) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2e-15) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else if (beta <= 5.6e+15) {
		tmp = (1.0 + beta) / (fma(beta, ((3.0 + beta) + 2.0), 6.0) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2e-15)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	elseif (beta <= 5.6e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(fma(beta, Float64(Float64(3.0 + beta) + 2.0), 6.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 3.0)) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2e-15], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 5.6e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta * N[(N[(3.0 + beta), $MachinePrecision] + 2.0), $MachinePrecision] + 6.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.1999999999999998e-15

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 92.3

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 92.3%

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 6.1999999999999998e-15 < beta < 5.6e15

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 97.6

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right)} \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \color{blue}{\left(\left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right) + 6\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\color{blue}{\left(\beta \cdot \left(3 + \beta\right) + 2 \cdot \beta\right)} + 6\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta \cdot \left(3 + \beta\right) + \color{blue}{\beta \cdot 2}\right) + 6\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\color{blue}{\beta \cdot \left(\left(3 + \beta\right) + 2\right)} + 6\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right)}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \color{blue}{\left(3 + \beta\right) + 2}, 6\right)} \]

      11. lower-+.f64 67.3

          \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \color{blue}{\left(3 + \beta\right)} + 2, 6\right)} \]

   9. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right)}} \]

   if 5.6e15 < beta

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\mathsf{neg}\left(\left(\alpha + 1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{-\left(\mathsf{neg}\left(\color{blue}{\left(1 + \alpha\right)}\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      7. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      9. unsub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      10. lower--.f64 83.0

          \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 83.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2e-15)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (if (<= beta 1.15e+16)
     (/ (+ 1.0 beta) (* (fma beta (+ (+ 3.0 beta) 2.0) 6.0) (+ 2.0 beta)))
     (/ (/ (- alpha -1.0) beta) (+ (+ alpha beta) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2e-15) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else if (beta <= 1.15e+16) {
		tmp = (1.0 + beta) / (fma(beta, ((3.0 + beta) + 2.0), 6.0) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / ((alpha + beta) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2e-15)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	elseif (beta <= 1.15e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(fma(beta, Float64(Float64(3.0 + beta) + 2.0), 6.0) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(Float64(alpha + beta) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2e-15], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.15e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta * N[(N[(3.0 + beta), $MachinePrecision] + 2.0), $MachinePrecision] + 6.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.1999999999999998e-15

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 92.3

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 92.3%

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 6.1999999999999998e-15 < beta < 1.15e16

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 97.6

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{\left(2 + \beta\right)} \cdot \left(6 + \left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \color{blue}{\left(\left(2 \cdot \beta + \beta \cdot \left(3 + \beta\right)\right) + 6\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\color{blue}{\left(\beta \cdot \left(3 + \beta\right) + 2 \cdot \beta\right)} + 6\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta \cdot \left(3 + \beta\right) + \color{blue}{\beta \cdot 2}\right) + 6\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\color{blue}{\beta \cdot \left(\left(3 + \beta\right) + 2\right)} + 6\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right)}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \color{blue}{\left(3 + \beta\right) + 2}, 6\right)} \]

      11. lower-+.f64 67.3

          \[\leadsto \frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \color{blue}{\left(3 + \beta\right)} + 2, 6\right)} \]

   9. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right)}} \]

   if 1.15e16 < beta

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\beta, \left(3 + \beta\right) + 2, 6\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.25}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 5.0) (/ 0.25 (+ 1.0 t_0)) (/ (/ (- alpha -1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 5.0d0) then
        tmp = 0.25d0 / (1.0d0 + t_0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 5.0) {
		tmp = 0.25 / (1.0 + t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 5.0:
		tmp = 0.25 / (1.0 + t_0)
	else:
		tmp = ((alpha - -1.0) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(0.25 / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = 0.25 / (1.0 + t_0);
	else
		tmp = ((alpha - -1.0) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5.0], N[(0.25 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 63.4%

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

   if 5 < beta

   1. Initial program 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (if (<= beta 1.35e+154)
     (/ (- alpha -1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else if (beta <= 1.35e+154) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 90.4

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.2999999999999998 < beta < 1.35000000000000003e154

   1. Initial program 89.4%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 70.7

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   if 1.35000000000000003e154 < beta

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 89.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.6%

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

   10. Applied rewrites 88.8%

       \[\leadsto \frac{\frac{\alpha}{\beta}}{\beta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ 0.25 (+ 1.0 (+ (+ alpha beta) 2.0)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 0.25d0 / (1.0d0 + ((alpha + beta) + 2.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(0.25 / Float64(1.0 + Float64(Float64(alpha + beta) + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(0.25 / N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 63.4%

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

   if 6.5 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 78.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.5%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 90.4

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.2999999999999998 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 78.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.5%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.3)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ (- alpha -1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = (alpha - -1.0) / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.3)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.3], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2999999999999998

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 90.4

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.2999999999999998 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 78.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 78.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.0], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 90.4

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 78.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.6%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 91.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (fma
    (fma -0.011574074074074073 alpha -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.0], N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 99.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 90.4

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 78.6

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.6%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 74.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2e+44)
   (fma
    (fma -0.011574074074074073 alpha -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2e+44) {
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2e+44)
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2e+44], N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000002e44

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

      18. metadata-eval 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 98.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

      12. lower-+.f64 85.0

          \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   9. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 2.0000000000000002e44 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

      4. lower-*.f64 85.4

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 54.7%

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

Alternative 18: 45.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma
  (fma -0.011574074074074073 alpha -0.027777777777777776)
  alpha
  0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   18. metadata-eval 93.7

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

6. Applied rewrites 93.7%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

7. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   5. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

   8. distribute-lft-out N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

   12. lower-+.f64 67.1

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

9. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

10. Taylor expanded in alpha around 0

    \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
11. Step-by-step derivation

12. Applied rewrites 41.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]
13. Add Preprocessing

Alternative 19: 45.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(-0.027777777777777776, \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma -0.027777777777777776 alpha 0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(-0.027777777777777776, alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(-0.027777777777777776, alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(-0.027777777777777776 * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   18. metadata-eval 93.7

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

6. Applied rewrites 93.7%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

7. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   5. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

   8. distribute-lft-out N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

   12. lower-+.f64 67.1

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

9. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

10. Taylor expanded in alpha around 0

    \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \alpha} \]
11. Step-by-step derivation

12. Applied rewrites 41.6%

    \[\leadsto \mathsf{fma}\left(-0.027777777777777776, \color{blue}{\alpha}, 0.08333333333333333\right) \]
13. Add Preprocessing

Alternative 20: 45.0% accurate, 84.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 2\right)}} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(3 + \left(\beta + \alpha\right)\right) + 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\beta + \alpha}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\color{blue}{\alpha + \beta}, 3 + \left(\beta + \alpha\right), 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\beta + \alpha\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \color{blue}{\left(\alpha + \beta\right)}, 2 \cdot \left(3 + \left(\beta + \alpha\right)\right)\right)} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(3 + \left(\beta + \alpha\right)\right)}\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), 2 \cdot \color{blue}{\left(\left(\beta + \alpha\right) + 3\right)}\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{2 \cdot \left(\beta + \alpha\right) + 2 \cdot 3}\right)} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \color{blue}{\mathsf{fma}\left(2, \beta + \alpha, 2 \cdot 3\right)}\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\beta + \alpha}, 2 \cdot 3\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \color{blue}{\alpha + \beta}, 2 \cdot 3\right)\right)} \]

   18. metadata-eval 93.7

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, \color{blue}{6}\right)\right)} \]

6. Applied rewrites 93.7%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\alpha + \beta, 3 + \left(\alpha + \beta\right), \mathsf{fma}\left(2, \alpha + \beta, 6\right)\right)}} \]

7. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{\left(2 + \alpha\right) \cdot \left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(6 + \left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right)\right) \cdot \left(2 + \alpha\right)}} \]

   5. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\left(\left(2 \cdot \alpha + \alpha \cdot \left(3 + \alpha\right)\right) + 6\right)} \cdot \left(2 + \alpha\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\left(\alpha \cdot \left(3 + \alpha\right) + 2 \cdot \alpha\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{1 + \alpha}{\left(\left(\alpha \cdot \left(3 + \alpha\right) + \color{blue}{\alpha \cdot 2}\right) + 6\right) \cdot \left(2 + \alpha\right)} \]

   8. distribute-lft-out N/A

      \[\leadsto \frac{1 + \alpha}{\left(\color{blue}{\alpha \cdot \left(\left(3 + \alpha\right) + 2\right)} + 6\right) \cdot \left(2 + \alpha\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right)} \cdot \left(2 + \alpha\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right) + 2}, 6\right) \cdot \left(2 + \alpha\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \color{blue}{\left(3 + \alpha\right)} + 2, 6\right) \cdot \left(2 + \alpha\right)} \]

   12. lower-+.f64 67.1

       \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

9. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\mathsf{fma}\left(\alpha, \left(3 + \alpha\right) + 2, 6\right) \cdot \left(2 + \alpha\right)}} \]

10. Taylor expanded in alpha around 0

    \[\leadsto \frac{1}{12} \]
11. Step-by-step derivation

12. Applied rewrites 41.9%

    \[\leadsto 0.08333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024327 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))