Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.5%

Time: 15.9s

Alternatives: 25

Speedup: 3.2×

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, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0}}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 4.2e+96)
     (/
      (/ (+ (fma alpha beta (+ beta alpha)) 1.0) t_0)
      (* t_0 (+ alpha (+ beta 3.0))))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4.2e+96) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / t_0) / (t_0 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 4.2e+96)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0))));
	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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4.2e+96], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.2000000000000002e96

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if 4.2000000000000002e96 < beta

   1. Initial program 79.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)} \cdot \left(1 + \left(\beta + \mathsf{fma}\left(\alpha, \beta, \alpha\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.3e+96)
     (*
      (/ 1.0 (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
      (+ 1.0 (+ beta (fma alpha beta alpha))))
     (/ (/ (+ 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 <= 3.3e+96) {
		tmp = (1.0 / ((alpha + (beta + 3.0)) * (t_0 * t_0))) * (1.0 + (beta + fma(alpha, beta, alpha)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.3e+96)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0))) * Float64(1.0 + Float64(beta + fma(alpha, beta, alpha))));
	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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.3e+96], N[(N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(beta + N[(alpha * beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.29999999999999984e96

   1. Initial program 98.8%

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

   4. Applied rewrites 98.7%

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

   6. Applied rewrites 90.8%

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

   if 3.29999999999999984e96 < beta

   1. Initial program 79.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(1 + \left(\beta + \mathsf{fma}\left(\alpha, \beta, \alpha\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+27)
     (/
      (+ (fma alpha beta (+ beta alpha)) 1.0)
      (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ 1.0 beta) (/ beta (+ alpha 1.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+27) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = (1.0 / beta) / (beta / (alpha + 1.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+27)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta / Float64(alpha + 1.0)));
	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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+27], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999979e27

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

   4. Applied rewrites 93.5%

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

   if 4.99999999999999979e27 < beta

   1. Initial program 82.2%

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

   3. 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 78.9

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

   5. Applied rewrites 78.9%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.1

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

   9. Applied rewrites 85.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e+14)
   (*
    (/ 1.0 (* (+ beta 2.0) (* (+ beta 2.0) (+ beta (+ alpha 3.0)))))
    (+ beta 1.0))
   (/ (/ 1.0 (+ alpha (+ beta 3.0))) (/ beta (+ alpha 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	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) :: tmp
    if (beta <= 3.2d+14) then
        tmp = (1.0d0 / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + (alpha + 3.0d0))))) * (beta + 1.0d0)
    else
        tmp = (1.0d0 / (alpha + (beta + 3.0d0))) / (beta / (alpha + 1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2e+14:
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0)
	else:
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e+14)
		tmp = Float64(Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + Float64(alpha + 3.0))))) * Float64(beta + 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(alpha + Float64(beta + 3.0))) / Float64(beta / Float64(alpha + 1.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2e+14)
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	else
		tmp = (1.0 / (alpha + (beta + 3.0))) / (beta / (alpha + 1.0));
	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, 3.2e+14], N[(N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e14

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*l/ N/A

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

      14. associate-/r/ N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 68.0%

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

   if 3.2e14 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 N/A

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

      13. clear-num N/A

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

      14. associate-*l/ N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 85.3

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

   7. Applied rewrites 85.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta}{\alpha + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 3.2e+14)
     (*
      (/ 1.0 (* (+ beta 2.0) (* (+ beta 2.0) (+ beta (+ alpha 3.0)))))
      (+ beta 1.0))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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 = (beta + alpha) + 2.0d0
    if (beta <= 3.2d+14) then
        tmp = (1.0d0 / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + (alpha + 3.0d0))))) * (beta + 1.0d0)
    else
        tmp = ((alpha + 1.0d0) / t_0) / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 3.2e+14:
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0)
	else:
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 3.2e+14)
		tmp = Float64(Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + Float64(alpha + 3.0))))) * Float64(beta + 1.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 3.2e+14)
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	else
		tmp = ((alpha + 1.0) / t_0) / (1.0 + 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 3.2e+14], N[(N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e14

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*l/ N/A

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

      14. associate-/r/ N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 68.0%

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

   if 3.2e14 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e+14)
   (*
    (/ 1.0 (* (+ beta 2.0) (* (+ beta 2.0) (+ beta (+ alpha 3.0)))))
    (+ beta 1.0))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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 <= 3.2d+14) then
        tmp = (1.0d0 / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + (alpha + 3.0d0))))) * (beta + 1.0d0)
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2e+14:
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0)
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e+14)
		tmp = Float64(Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + Float64(alpha + 3.0))))) * Float64(beta + 1.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2e+14)
		tmp = (1.0 / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))) * (beta + 1.0);
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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, 3.2e+14], N[(N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e14

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*l/ N/A

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

      14. associate-/r/ N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 68.0%

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

   if 3.2e14 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)} \cdot \left(\beta + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e+14)
   (/ (+ beta 1.0) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta (+ alpha 3.0)))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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 <= 3.2d+14) then
        tmp = (beta + 1.0d0) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + (alpha + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2e+14:
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e+14)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + Float64(alpha + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2e+14)
		tmp = (beta + 1.0) / ((beta + 2.0) * ((beta + 2.0) * (beta + (alpha + 3.0))));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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, 3.2e+14], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e14

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*l/ N/A

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

      14. associate-/r/ N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 68.0%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. associate-*l/ N/A

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

       9. *-lft-identity N/A

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

       10. lower-/.f64 68.0

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

   11. Applied rewrites 68.0%

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

   if 3.2e14 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e+14)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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 <= 3.2d+14) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 3.0d0))) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e+14) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.2e+14:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e+14)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.2e+14)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / 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, 3.2e+14], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e14

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 3.2e14 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\alpha + 3, \beta, 4\right), \mathsf{fma}\left(\alpha, 4, 12\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ 1.0 (fma beta (fma (+ alpha 3.0) beta 4.0) (fma alpha 4.0 12.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 1.0 / fma(beta, fma((alpha + 3.0), beta, 4.0), fma(alpha, 4.0, 12.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(1.0 / fma(beta, fma(Float64(alpha + 3.0), beta, 4.0), fma(alpha, 4.0, 12.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / 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, 4.8], N[(1.0 / N[(beta * N[(N[(alpha + 3.0), $MachinePrecision] * beta + 4.0), $MachinePrecision] + N[(alpha * 4.0 + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(3 + \alpha\right) + \beta \cdot \left(4 + \beta \cdot \left(3 + \alpha\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \left(4 + \beta \cdot \left(3 + \alpha\right)\right) + 4 \cdot \left(3 + \alpha\right)}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, 4 + \beta \cdot \left(3 + \alpha\right), 4 \cdot \left(3 + \alpha\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(3 + \alpha\right) + 4}, 4 \cdot \left(3 + \alpha\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\left(3 + \alpha\right) \cdot \beta} + 4, 4 \cdot \left(3 + \alpha\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(3 + \alpha, \beta, 4\right)}, 4 \cdot \left(3 + \alpha\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\color{blue}{3 + \alpha}, \beta, 4\right), 4 \cdot \left(3 + \alpha\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), 4 \cdot \color{blue}{\left(\alpha + 3\right)}\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \color{blue}{\alpha \cdot 4 + 3 \cdot 4}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \alpha \cdot 4 + \color{blue}{12}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \alpha \cdot 4 + \color{blue}{4 \cdot 3}\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \color{blue}{\mathsf{fma}\left(\alpha, 4, 4 \cdot 3\right)}\right)} \]

      12. metadata-eval 68.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \mathsf{fma}\left(\alpha, 4, \color{blue}{12}\right)\right)} \]

   10. Applied rewrites 68.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(3 + \alpha, \beta, 4\right), \mathsf{fma}\left(\alpha, 4, 12\right)\right)}} \]

   if 4.79999999999999982 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 80.2

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

   7. Applied rewrites 80.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\alpha + 3, \beta, 4\right), \mathsf{fma}\left(\alpha, 4, 12\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.2)
   (/ 1.0 (* 4.0 (+ beta (+ alpha 3.0))))
   (if (<= beta 2.05e+154)
     (/ (+ alpha 1.0) (* beta (+ alpha (+ beta 3.0))))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) / (beta * (alpha + (beta + 3.0)));
	} else {
		tmp = (alpha / 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 <= 4.2d0) then
        tmp = 1.0d0 / (4.0d0 * (beta + (alpha + 3.0d0)))
    else if (beta <= 2.05d+154) then
        tmp = (alpha + 1.0d0) / (beta * (alpha + (beta + 3.0d0)))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.2) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) / (beta * (alpha + (beta + 3.0)));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.2:
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)))
	elif beta <= 2.05e+154:
		tmp = (alpha + 1.0) / (beta * (alpha + (beta + 3.0)))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(1.0 / Float64(4.0 * Float64(beta + Float64(alpha + 3.0))));
	elseif (beta <= 2.05e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	elseif (beta <= 2.05e+154)
		tmp = (alpha + 1.0) / (beta * (alpha + (beta + 3.0)));
	else
		tmp = (alpha / 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, 4.2], N[(1.0 / N[(4.0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.05e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.20000000000000018

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \beta + 4 \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

      4. lower-+.f64 68.4

         \[\leadsto \frac{1}{4 \cdot \left(\beta + \color{blue}{\left(3 + \alpha\right)}\right)} \]

   10. Applied rewrites 68.4%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

   if 4.20000000000000018 < beta < 2.05e154

   1. Initial program 93.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 64.2

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

   5. Applied rewrites 64.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 2.05e154 < beta

   1. Initial program 76.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 94.3

         \[\leadsto \frac{\color{blue}{\frac{\alpha}{\beta}}}{\beta} \]

   10. Applied rewrites 94.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.15:\\ \;\;\;\;\frac{1}{t\_0 \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\beta}\\ \end{array} \end{array} \]

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 3.0))))
   (if (<= beta 3.15)
     (/ 1.0 (* t_0 (fma beta beta 4.0)))
     (/ (/ (+ alpha 1.0) t_0) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.15) {
		tmp = 1.0 / (t_0 * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 3.15)
		tmp = Float64(1.0 / Float64(t_0 * fma(beta, beta, 4.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_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[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.15], N[(1.0 / N[(t$95$0 * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.14999999999999991

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\left(4 + {\beta}^{2}\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({\beta}^{2} + 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\beta \cdot \beta} + 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. lower-fma.f64 68.7

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   10. Applied rewrites 68.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 3.14999999999999991 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-+l+ N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 80.2

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

   7. Applied rewrites 80.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.15:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 1.0 (* 4.0 (+ beta (+ alpha 3.0))))
   (if (<= beta 2.05e+154)
     (* (+ alpha 1.0) (/ 1.0 (* beta beta)))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	} else {
		tmp = (alpha / 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 <= 6.4d0) then
        tmp = 1.0d0 / (4.0d0 * (beta + (alpha + 3.0d0)))
    else if (beta <= 2.05d+154) then
        tmp = (alpha + 1.0d0) * (1.0d0 / (beta * beta))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)))
	elif beta <= 2.05e+154:
		tmp = (alpha + 1.0) * (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 <= 6.4)
		tmp = Float64(1.0 / Float64(4.0 * Float64(beta + Float64(alpha + 3.0))));
	elseif (beta <= 2.05e+154)
		tmp = Float64(Float64(alpha + 1.0) * Float64(1.0 / Float64(beta * beta)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	elseif (beta <= 2.05e+154)
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	else
		tmp = (alpha / 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, 6.4], N[(1.0 / N[(4.0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.05e+154], N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.4000000000000004

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \beta + 4 \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

      4. lower-+.f64 68.4

         \[\leadsto \frac{1}{4 \cdot \left(\beta + \color{blue}{\left(3 + \alpha\right)}\right)} \]

   10. Applied rewrites 68.4%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

   if 6.4000000000000004 < beta < 2.05e154

   1. Initial program 93.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.9

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

   7. Applied rewrites 63.9%

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

   if 2.05e154 < beta

   1. Initial program 76.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 94.3

         \[\leadsto \frac{\color{blue}{\frac{\alpha}{\beta}}}{\beta} \]

   10. Applied rewrites 94.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 1.0 (* 4.0 (+ beta (+ alpha 3.0))))
   (if (<= beta 2.05e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / 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 <= 6.4d0) then
        tmp = 1.0d0 / (4.0d0 * (beta + (alpha + 3.0d0)))
    else if (beta <= 2.05d+154) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else if (beta <= 2.05e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)))
	elif beta <= 2.05e+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 <= 6.4)
		tmp = Float64(1.0 / Float64(4.0 * Float64(beta + Float64(alpha + 3.0))));
	elseif (beta <= 2.05e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	elseif (beta <= 2.05e+154)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / 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, 6.4], N[(1.0 / N[(4.0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.05e+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 < 6.4000000000000004

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \beta + 4 \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

      4. lower-+.f64 68.4

         \[\leadsto \frac{1}{4 \cdot \left(\beta + \color{blue}{\left(3 + \alpha\right)}\right)} \]

   10. Applied rewrites 68.4%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

   if 6.4000000000000004 < beta < 2.05e154

   1. Initial program 93.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   if 2.05e154 < beta

   1. Initial program 76.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 94.3

         \[\leadsto \frac{\color{blue}{\frac{\alpha}{\beta}}}{\beta} \]

   10. Applied rewrites 94.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) (fma beta beta 4.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * fma(beta, beta, 4.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	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, 3.9], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\left(4 + {\beta}^{2}\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({\beta}^{2} + 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\beta \cdot \beta} + 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. lower-fma.f64 68.7

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   10. Applied rewrites 68.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 3.89999999999999991 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta + 3}} \]

      2. lower-+.f64 80.0

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta + 3}} \]

   8. Applied rewrites 80.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) (fma beta beta 4.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * fma(beta, beta, 4.0)));
	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, 4.5], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\left(4 + {\beta}^{2}\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({\beta}^{2} + 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\beta \cdot \beta} + 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. lower-fma.f64 68.7

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   10. Applied rewrites 68.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\beta, \beta, 4\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 4.5 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 79.9

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

   7. Applied rewrites 79.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 1.0 (* 4.0 (+ beta (+ alpha 3.0))))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} 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 <= 6.4d0) then
        tmp = 1.0d0 / (4.0d0 * (beta + (alpha + 3.0d0)))
    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 <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(1.0 / Float64(4.0 * Float64(beta + Float64(alpha + 3.0))));
	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 <= 6.4)
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	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, 6.4], N[(1.0 / N[(4.0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \beta + 4 \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

      4. lower-+.f64 68.4

         \[\leadsto \frac{1}{4 \cdot \left(\beta + \color{blue}{\left(3 + \alpha\right)}\right)} \]

   10. Applied rewrites 68.4%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

   if 6.4000000000000004 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 79.9

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

   7. Applied rewrites 79.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 94.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.4)
   (/ 1.0 (* 4.0 (+ beta (+ alpha 3.0))))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} 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 <= 6.4d0) then
        tmp = 1.0d0 / (4.0d0 * (beta + (alpha + 3.0d0)))
    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 <= 6.4) {
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.4:
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.4)
		tmp = Float64(1.0 / Float64(4.0 * Float64(beta + Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.4)
		tmp = 1.0 / (4.0 * (beta + (alpha + 3.0)));
	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, 6.4], N[(1.0 / N[(4.0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4000000000000004

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \beta + 4 \cdot \left(3 + \alpha\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

      4. lower-+.f64 68.4

         \[\leadsto \frac{1}{4 \cdot \left(\beta + \color{blue}{\left(3 + \alpha\right)}\right)} \]

   10. Applied rewrites 68.4%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot \left(\beta + \left(3 + \alpha\right)\right)}} \]

   if 6.4000000000000004 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{1}{4 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 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 <= 3.4)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 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, 3.4], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, \frac{2}{81} \cdot \alpha - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if 3.39999999999999991 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	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, 2.2], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, \frac{2}{81} \cdot \alpha - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if 2.2000000000000002 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 72.6

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

   8. Applied rewrites 72.6%

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

Alternative 20: 91.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.3) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 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 <= 3.3)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 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, 3.3], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / 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. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, \frac{2}{81} \cdot \alpha - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if 3.2999999999999998 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

Alternative 21: 91.8% accurate, 3.6× speedup

Math

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

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
    alpha
    (fma alpha -0.011574074074074073 -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 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(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 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[(alpha * N[(alpha * -0.011574074074074073 + -0.027777777777777776), $MachinePrecision] + 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. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{-5}{432} \cdot \alpha - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{-5}{432} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{-5}{432}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{-5}{432} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      6. lower-fma.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if 3 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

Alternative 22: 47.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (fma
    alpha
    (fma alpha -0.011574074074074073 -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(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_] := If[LessEqual[beta, 9.0], N[(alpha * N[(alpha * -0.011574074074074073 + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{-5}{432} \cdot \alpha - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{-5}{432} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{-5}{432}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{-5}{432} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      6. lower-fma.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if 9 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 6.7

         \[\leadsto \color{blue}{\frac{1}{\beta}} \]

   8. Applied rewrites 6.7%

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

Alternative 23: 47.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (fma alpha -0.027777777777777776 0.08333333333333333)
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = fma(alpha, -0.027777777777777776, 0.08333333333333333);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = fma(alpha, -0.027777777777777776, 0.08333333333333333);
	else
		tmp = Float64(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_] := If[LessEqual[beta, 9.0], N[(alpha * -0.027777777777777776 + 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]

      2. lower-+.f64 67.3

         \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

   10. Applied rewrites 67.3%

       \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

   11. Taylor expanded in alpha around 0

       \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{-1}{36} \cdot \alpha + \frac{1}{12}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\alpha \cdot \frac{-1}{36}} + \frac{1}{12} \]

       3. lower-fma.f64 65.6

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

   13. Applied rewrites 65.6%

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

   if 9 < beta

   1. Initial program 84.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 6.7

         \[\leadsto \color{blue}{\frac{1}{\beta}} \]

   8. Applied rewrites 6.7%

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

Alternative 24: 45.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma alpha -0.027777777777777776 0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(alpha, -0.027777777777777776, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(alpha, -0.027777777777777776, 0.08333333333333333)
end

Wolfram

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

Derivation

1. Initial program 94.4%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-+.f64 71.8

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]

   2. lower-+.f64 45.1

      \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

10. Applied rewrites 45.1%

    \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

11. Taylor expanded in alpha around 0

    \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \alpha + \frac{1}{12}} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\alpha \cdot \frac{-1}{36}} + \frac{1}{12} \]

    3. lower-fma.f64 43.5

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

13. Applied rewrites 43.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)} \]
14. Add Preprocessing

Alternative 25: 45.2% 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 94.4%

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-+.f64 71.8

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]

   2. lower-+.f64 45.1

      \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

10. Applied rewrites 45.1%

    \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

11. Taylor expanded in alpha around 0

    \[\leadsto \color{blue}{\frac{1}{12}} \]
12. Step-by-step derivation

13. Applied rewrites 43.9%

    \[\leadsto \color{blue}{0.08333333333333333} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))