Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.6%

Time: 11.1s

Alternatives: 19

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(2 + \left(\beta + \alpha\right)\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \left(1 + \alpha\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))))
   (if (<= alpha 2.6e-16)
     (/
      (*
       (pow (+ 2.0 (+ beta alpha)) -2.0)
       (+ 1.0 (fma beta alpha (+ beta alpha))))
      t_0)
     (/
      (/
       (-
        (+ (+ (/ 1.0 beta) (/ alpha beta)) (+ 1.0 alpha))
        (* (/ (fma 2.0 alpha 4.0) beta) (+ 1.0 alpha)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double tmp;
	if (alpha <= 2.6e-16) {
		tmp = (pow((2.0 + (beta + alpha)), -2.0) * (1.0 + fma(beta, alpha, (beta + alpha)))) / t_0;
	} else {
		tmp = (((((1.0 / beta) + (alpha / beta)) + (1.0 + alpha)) - ((fma(2.0, alpha, 4.0) / beta) * (1.0 + alpha))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (alpha <= 2.6e-16)
		tmp = Float64(Float64((Float64(2.0 + Float64(beta + alpha)) ^ -2.0) * Float64(1.0 + fma(beta, alpha, Float64(beta + alpha)))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) + Float64(1.0 + alpha)) - Float64(Float64(fma(2.0, alpha, 4.0) / beta) * Float64(1.0 + alpha))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.6e-16], N[(N[(N[Power[N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.5999999999999998e-16

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

   if 2.5999999999999998e-16 < alpha

   1. Initial program 86.6%

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 19.0

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

   7. Applied rewrites 19.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + 1\right) + \left(\frac{\alpha}{\beta} + \frac{1}{\beta}\right)\right) - \left(\alpha + 1\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(2 + \left(\beta + \alpha\right)\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)\right)}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \left(1 + \alpha\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\alpha \leq 6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_0} \cdot \frac{-1}{t\_0}}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \left(1 + \alpha\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= alpha 6e-39)
     (/
      (* (/ (- -1.0 (fma beta alpha (+ beta alpha))) t_0) (/ -1.0 t_0))
      (+ t_0 1.0))
     (/
      (/
       (-
        (+ (+ (/ 1.0 beta) (/ alpha beta)) (+ 1.0 alpha))
        (* (/ (fma 2.0 alpha 4.0) beta) (+ 1.0 alpha)))
       beta)
      (+ 3.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (alpha <= 6e-39) {
		tmp = (((-1.0 - fma(beta, alpha, (beta + alpha))) / t_0) * (-1.0 / t_0)) / (t_0 + 1.0);
	} else {
		tmp = (((((1.0 / beta) + (alpha / beta)) + (1.0 + alpha)) - ((fma(2.0, alpha, 4.0) / beta) * (1.0 + alpha))) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (alpha <= 6e-39)
		tmp = Float64(Float64(Float64(Float64(-1.0 - fma(beta, alpha, Float64(beta + alpha))) / t_0) * Float64(-1.0 / t_0)) / Float64(t_0 + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) + Float64(1.0 + alpha)) - Float64(Float64(fma(2.0, alpha, 4.0) / beta) * Float64(1.0 + alpha))) / beta) / Float64(3.0 + Float64(beta + alpha)));
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 6e-39], N[(N[(N[(N[(-1.0 - N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.00000000000000055e-39

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if 6.00000000000000055e-39 < alpha

   1. Initial program 87.3%

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 20.2

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

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\alpha + 1\right) + \left(\frac{\alpha}{\beta} + \frac{1}{\beta}\right)\right) - \left(\alpha + 1\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}}{3 + \left(\beta + \alpha\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)} \cdot \frac{-1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right) + \left(1 + \alpha\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+138}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_0} \cdot \frac{-1}{t\_0}}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1e+138)
     (/
      (* (/ (- -1.0 (fma beta alpha (+ beta alpha))) t_0) (/ -1.0 t_0))
      (+ t_0 1.0))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+138) {
		tmp = (((-1.0 - fma(beta, alpha, (beta + alpha))) / t_0) * (-1.0 / t_0)) / (t_0 + 1.0);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1e+138)
		tmp = Float64(Float64(Float64(Float64(-1.0 - fma(beta, alpha, Float64(beta + alpha))) / t_0) * Float64(-1.0 / t_0)) / Float64(t_0 + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+138], N[(N[(N[(N[(-1.0 - N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e138

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 98.4%

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

   if 1e138 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 90.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+138}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)} \cdot \frac{-1}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)} \cdot \left(t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2.1e+17)
     (/ 1.0 (* (/ t_1 (+ 1.0 (fma beta alpha (+ beta alpha)))) (* t_0 t_1)))
     (/ (/ (+ 1.0 alpha) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.1e+17) {
		tmp = 1.0 / ((t_1 / (1.0 + fma(beta, alpha, (beta + alpha)))) * (t_0 * t_1));
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.1e+17)
		tmp = Float64(1.0 / Float64(Float64(t_1 / Float64(1.0 + fma(beta, alpha, Float64(beta + alpha)))) * Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1);
	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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.1e+17], N[(1.0 / N[(N[(t$95$1 / N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.1e17

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if 2.1e17 < beta

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 83.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)} \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_1}}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2e+17)
     (/ (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_1) (* t_0 t_1))
     (/ (/ (+ 1.0 alpha) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2e+17) {
		tmp = ((1.0 + fma(beta, alpha, (beta + alpha))) / t_1) / (t_0 * t_1);
	} else {
		tmp = ((1.0 + alpha) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2e+17)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_1) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / t_1);
	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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e17

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if 2e17 < beta

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 83.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(t\_1 \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ 3.0 (+ beta alpha))))
   (if (<= beta 2e+17)
     (/ (+ 1.0 (fma beta alpha (+ beta alpha))) (* (* t_1 t_0) t_0))
     (/ (/ (+ 1.0 alpha) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 2e+17) {
		tmp = (1.0 + fma(beta, alpha, (beta + alpha))) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2e+17)
		tmp = Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / Float64(Float64(t_1 * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e17

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.8%

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

   if 2e17 < beta

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 83.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2.35e+16)
     (/ (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 3.0 beta))) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.35e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / 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 = 2.0d0 + (beta + alpha)
    if (beta <= 2.35d+16) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (3.0d0 + beta))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + alpha))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.35e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 2.35e+16:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.35e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(2.0 + beta) * Float64(3.0 + beta))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 2.35e+16)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.35e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.35e16

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.9

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

   5. Applied rewrites 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.9%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.9

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

   10. Applied rewrites 72.9%

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

   if 2.35e16 < beta

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 83.3

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

   5. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 83.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 5.4)
     (/ (/ (+ 1.0 alpha) (* (+ 2.0 alpha) (+ 3.0 alpha))) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5.4) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / 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 = 2.0d0 + (beta + alpha)
    if (beta <= 5.4d0) then
        tmp = ((1.0d0 + alpha) / ((2.0d0 + alpha) * (3.0d0 + alpha))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + alpha))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 5.4) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 5.4:
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / t_0
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + alpha) * Float64(3.0 + alpha))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 5.4)
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / t_0;
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.4], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if 5.4000000000000004 < beta

   1. Initial program 86.3%

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

   3. 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 82.4

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

   5. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 82.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ (/ (+ 1.0 alpha) (* (+ 2.0 alpha) (+ 3.0 alpha))) (+ 2.0 alpha))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / (2.0 + alpha);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / (2.0 + (beta + alpha));
	}
	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 <= 3.8d0) then
        tmp = ((1.0d0 + alpha) / ((2.0d0 + alpha) * (3.0d0 + alpha))) / (2.0d0 + alpha)
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (beta + alpha))) / (2.0d0 + (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / (2.0 + alpha);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / (2.0 + alpha)
	else:
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / (2.0 + (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(2.0 + alpha) * Float64(3.0 + alpha))) / Float64(2.0 + alpha));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = ((1.0 + alpha) / ((2.0 + alpha) * (3.0 + alpha))) / (2.0 + alpha);
	else
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / (2.0 + (beta + alpha));
	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, 3.8], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 99.0

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

   13. Applied rewrites 99.0%

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

   if 3.7999999999999998 < beta

   1. Initial program 86.3%

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

   3. 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 82.4

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

   5. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 82.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 2.6)
     (/ (fma 0.027777777777777776 alpha 0.16666666666666666) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 2.6) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.6], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1}{6} + \color{blue}{\frac{1}{36} \cdot \alpha}}{2 + \left(\alpha + \beta\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, \color{blue}{\alpha}, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)} \]

   if 2.60000000000000009 < beta

   1. Initial program 86.3%

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

   3. 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 82.4

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

   5. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 82.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \beta}\\ \end{array} \end{array} \]

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 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ beta alpha)))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (beta + alpha));
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / (2.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / Float64(2.0 + 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, 3.0], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1}{6} + \color{blue}{\frac{1}{36} \cdot \alpha}}{2 + \left(\alpha + \beta\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, \color{blue}{\alpha}, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)} \]

   if 3 < beta

   1. Initial program 86.3%

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

   3. 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 82.4

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

   5. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.9

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

   10. Applied rewrites 81.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{2 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ beta alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (beta + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(beta + alpha)));
	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, 5.2], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1}{6} + \color{blue}{\frac{1}{36} \cdot \alpha}}{2 + \left(\alpha + \beta\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, \color{blue}{\alpha}, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)} \]

   if 5.20000000000000018 < beta

   1. Initial program 86.3%

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 81.9

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

   7. Applied rewrites 81.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ beta alpha)))
   (if (<= beta 4e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (beta + alpha));
	} else if (beta <= 4e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(beta + alpha)));
	elseif (beta <= 4e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1}{6} + \color{blue}{\frac{1}{36} \cdot \alpha}}{2 + \left(\alpha + \beta\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, \color{blue}{\alpha}, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)} \]

   if 7.79999999999999982 < beta < 4.00000000000000015e154

   1. Initial program 88.7%

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

   3. 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 67.8

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

   5. Applied rewrites 67.8%

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

   if 4.00000000000000015e154 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.1%

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

   10. Applied rewrites 89.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (if (<= beta 4e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else if (beta <= 4e+154) {
		tmp = (1.0 + alpha) / (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 <= 8.0d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + alpha))
    else if (beta <= 4d+154) then
        tmp = (1.0d0 + alpha) / (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 <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else if (beta <= 4e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha))
	elif beta <= 4e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	elseif (beta <= 4e+154)
		tmp = Float64(Float64(1.0 + alpha) / 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 <= 8.0)
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	elseif (beta <= 4e+154)
		tmp = (1.0 + alpha) / (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, 8.0], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 72.3%

       \[\leadsto \frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)} \]

   if 8 < beta < 4.00000000000000015e154

   1. Initial program 88.7%

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

   3. 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 67.8

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

   5. Applied rewrites 67.8%

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

   if 4.00000000000000015e154 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.1%

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

   10. Applied rewrites 89.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ beta alpha)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (beta + alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.8], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.79999999999999982

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\frac{1}{6} + \color{blue}{\frac{1}{36} \cdot \alpha}}{2 + \left(\alpha + \beta\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, \color{blue}{\alpha}, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)} \]

   if 7.79999999999999982 < beta

   1. Initial program 86.3%

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

   3. 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 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 81.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 93.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = (1.0 + 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 <= 8.0d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + alpha))
    else
        tmp = (1.0d0 + 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 <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(1.0 + alpha) / 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 <= 8.0)
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	else
		tmp = (1.0 + 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, 8.0], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 72.3%

       \[\leadsto \frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)} \]

   if 8 < beta

   1. Initial program 86.3%

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

   3. 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 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 91.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.16666666666666666 (+ 2.0 (+ beta alpha)))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = 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 <= 8.0d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (beta + alpha))
    else
        tmp = 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 <= 8.0) {
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha))
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(beta + alpha)));
	else
		tmp = Float64(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 <= 8.0)
		tmp = 0.16666666666666666 / (2.0 + (beta + alpha));
	else
		tmp = 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, 8.0], N[(0.16666666666666666 / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 72.3%

       \[\leadsto \frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)} \]

   if 8 < beta

   1. Initial program 86.3%

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

   3. 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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.0:
		tmp = 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 (alpha <= 1.0)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 31.1

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

   5. Applied rewrites 31.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 30.3%

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

   if 1 < alpha

   1. Initial program 85.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 18.7

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 17.8%

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

Alternative 19: 31.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ alpha (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

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 = alpha / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return alpha / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return alpha / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(alpha / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = alpha / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. 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 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 18.8%

   \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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)))