Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.6%

Time: 11.0s

Alternatives: 20

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 100000000000:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_0}}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 100000000000.0)
     (/
      (/
       (/ (+ 1.0 (fma beta alpha (+ alpha beta))) t_0)
       (+ 3.0 (+ alpha beta)))
      t_0)
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (fma 2.0 alpha 5.0) beta) (+ 1.0 alpha)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 100000000000.0) {
		tmp = (((1.0 + fma(beta, alpha, (alpha + beta))) / t_0) / (3.0 + (alpha + beta))) / t_0;
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - ((fma(2.0, alpha, 5.0) / beta) * (1.0 + alpha))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 100000000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_0) / Float64(3.0 + Float64(alpha + beta))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(fma(2.0, alpha, 5.0) / 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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 100000000000.0], N[(N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * alpha + 5.0), $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 beta < 1e11

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if 1e11 < beta

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 91.5

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

   7. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(1 + \left(\alpha + \frac{1}{\beta}\right)\right) + \frac{\alpha}{\beta}\right) - \left(\alpha + 1\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}}{\left(\beta + \alpha\right) + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 100000000000:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta} \cdot \left(1 + \alpha\right)}{\beta}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \frac{\frac{\frac{\left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{0.16666666666666666}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta)))
        (t_1
         (/
          (/ (/ (+ (+ (* alpha beta) (+ alpha beta)) 1.0) t_0) t_0)
          (+ t_0 1.0))))
   (if (<= t_1 2e-6)
     (/ (+ 1.0 alpha) (* beta beta))
     (if (<= t_1 0.1) (/ 0.16666666666666666 t_0) (/ (/ alpha beta) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = (((((alpha * beta) + (alpha + beta)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	double tmp;
	if (t_1 <= 2e-6) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else if (t_1 <= 0.1) {
		tmp = 0.16666666666666666 / t_0;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = (((((alpha * beta) + (alpha + beta)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
    if (t_1 <= 2d-6) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else if (t_1 <= 0.1d0) then
        tmp = 0.16666666666666666d0 / t_0
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = (((((alpha * beta) + (alpha + beta)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	double tmp;
	if (t_1 <= 2e-6) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else if (t_1 <= 0.1) {
		tmp = 0.16666666666666666 / t_0;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = (((((alpha * beta) + (alpha + beta)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)
	tmp = 0
	if t_1 <= 2e-6:
		tmp = (1.0 + alpha) / (beta * beta)
	elif t_1 <= 0.1:
		tmp = 0.16666666666666666 / t_0
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha * beta) + Float64(alpha + beta)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
	tmp = 0.0
	if (t_1 <= 2e-6)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	elseif (t_1 <= 0.1)
		tmp = Float64(0.16666666666666666 / t_0);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = (((((alpha * beta) + (alpha + beta)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
	tmp = 0.0;
	if (t_1 <= 2e-6)
		tmp = (1.0 + alpha) / (beta * beta);
	elseif (t_1 <= 0.1)
		tmp = 0.16666666666666666 / t_0;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-6], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(0.16666666666666666 / t$95$0), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 1.99999999999999991e-6

   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 53.3

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

   5. Applied rewrites 53.3%

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

   if 1.99999999999999991e-6 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 95.7%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 1.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 51.0%

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

   10. Applied rewrites 54.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right) + 1}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right) + 1}{2 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1} \leq 0.1:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 10^{+59}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{t\_1}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

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 (+ alpha beta))) (t_1 (+ 2.0 (+ alpha beta))))
   (if (<= beta 1e+59)
     (/ (/ (/ (+ 1.0 (fma beta alpha (+ alpha beta))) 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 + (alpha + beta);
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1e+59) {
		tmp = (((1.0 + fma(beta, alpha, (alpha + beta))) / 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(alpha + beta))
	t_1 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1e+59)
		tmp = Float64(Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(alpha + beta))) / t_1) / 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+59], N[(N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $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 < 9.99999999999999972e58

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if 9.99999999999999972e58 < beta

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+59}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $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 99.5%

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

   if 2e17 < beta

   1. Initial program 83.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 97.0%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(1.0 + N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $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 95.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 2e17 < beta

   1. Initial program 83.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 94.1%

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

Alternative 6: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 2d+17) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (((3.0d0 + (alpha + beta)) * t_0) * t_0)
    else
        tmp = ((1.0d0 + alpha) / t_0) / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 2e+17:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (((3.0 + (alpha + beta)) * t_0) * t_0)
	else:
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0)
	return tmp

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 2e+17)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (((3.0 + (alpha + beta)) * t_0) * t_0);
	else
		tmp = ((1.0 + alpha) / t_0) / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $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 95.3%

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

   5. Taylor expanded in beta around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if 2e17 < beta

   1. Initial program 83.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 94.1%

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

Alternative 7: 99.5% accurate, 1.5× speedup

Math

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

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 (+ alpha beta))) (t_1 (+ 3.0 (+ alpha beta))))
   (if (<= beta 2e+17)
     (/ (* (+ 1.0 beta) (+ 1.0 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 + (alpha + beta);
	double t_1 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 2e+17) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = 3.0d0 + (alpha + beta)
    if (beta <= 2d+17) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((t_1 * t_0) * t_0)
    else
        tmp = ((1.0d0 + alpha) / t_1) / 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 + (alpha + beta);
	double t_1 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 2e+17) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 2e+17:
		tmp = ((1.0 + beta) * (1.0 + 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(alpha + beta))
	t_1 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2e+17)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + 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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = 3.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 2e+17)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	else
		tmp = ((1.0 + alpha) / t_1) / 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+17], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $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 95.3%

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

   5. Taylor expanded in beta around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if 2e17 < beta

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 94.1%

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

Alternative 8: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (+ alpha beta))))
   (if (<= beta 1.8e+16)
     (/ (/ (+ 1.0 beta) (* (+ 2.0 beta) (+ 3.0 beta))) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.8e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    if (beta <= 1.8d+16) then
        tmp = ((1.0d0 + beta) / ((2.0d0 + beta) * (3.0d0 + beta))) / t_0
    else
        tmp = ((1.0d0 + alpha) / (3.0d0 + (alpha + beta))) / 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 + (alpha + beta);
	double tmp;
	if (beta <= 1.8e+16) {
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 1.8e+16:
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0
	else:
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.8e+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(alpha + beta))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 1.8e+16)
		tmp = ((1.0 + beta) / ((2.0 + beta) * (3.0 + beta))) / t_0;
	else
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.8e+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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8e16

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 68.9

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

   7. Applied rewrites 68.9%

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

   if 1.8e16 < beta

   1. Initial program 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\mathsf{fma}\left(5 + \alpha, \alpha, 6\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (+ alpha beta))))
   (if (<= beta 6.0)
     (/ (/ (+ 1.0 alpha) (fma (+ 5.0 alpha) alpha 6.0)) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 6.0) {
		tmp = ((1.0 + alpha) / fma((5.0 + alpha), alpha, 6.0)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(Float64(Float64(1.0 + alpha) / fma(Float64(5.0 + alpha), alpha, 6.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.0], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(5.0 + alpha), $MachinePrecision] * alpha + 6.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 97.1%

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

   if 6 < beta

   1. Initial program 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 94.4%

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

Alternative 10: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (+ alpha beta))))
   (if (<= beta 2.45)
     (/
      (fma
       (fma
        (fma 0.03780864197530864 alpha -0.05092592592592592)
        alpha
        0.027777777777777776)
       alpha
       0.16666666666666666)
      t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ alpha beta))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 2.45) {
		tmp = fma(fma(fma(0.03780864197530864, alpha, -0.05092592592592592), alpha, 0.027777777777777776), alpha, 0.16666666666666666) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 2.45)
		tmp = Float64(fma(fma(fma(0.03780864197530864, alpha, -0.05092592592592592), alpha, 0.027777777777777776), alpha, 0.16666666666666666) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.45], N[(N[(N[(N[(0.03780864197530864 * alpha + -0.05092592592592592), $MachinePrecision] * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.4500000000000002

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{6} + \color{blue}{\alpha \cdot \left(\frac{1}{36} + \alpha \cdot \left(\frac{49}{1296} \cdot \alpha - \frac{11}{216}\right)\right)}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \color{blue}{\alpha}, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2} \]

   if 2.4500000000000002 < beta

   1. Initial program 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.0)
   (/
    (fma
     (fma
      (fma 0.03780864197530864 alpha -0.05092592592592592)
      alpha
      0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.0) {
		tmp = fma(fma(fma(0.03780864197530864, alpha, -0.05092592592592592), alpha, 0.027777777777777776), alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(fma(fma(fma(0.03780864197530864, alpha, -0.05092592592592592), alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + 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, 5.0], N[(N[(N[(N[(0.03780864197530864 * alpha + -0.05092592592592592), $MachinePrecision] * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{6} + \color{blue}{\alpha \cdot \left(\frac{1}{36} + \alpha \cdot \left(\frac{49}{1296} \cdot \alpha - \frac{11}{216}\right)\right)}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \color{blue}{\alpha}, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2} \]

   if 5 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 88.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \alpha, -0.05092592592592592\right), \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/
    (fma
     (fma -0.05092592592592592 alpha 0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{6} + \color{blue}{\alpha \cdot \left(\frac{1}{36} + \frac{-11}{216} \cdot \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \color{blue}{\alpha}, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2} \]

   if 4.79999999999999982 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 88.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/
    (fma
     (fma -0.05092592592592592 alpha 0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{6} + \color{blue}{\alpha \cdot \left(\frac{1}{36} + \frac{-11}{216} \cdot \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \color{blue}{\alpha}, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2} \]

   if 4.79999999999999982 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-+.f64 88.3

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

   8. Applied rewrites 88.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.7)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (if (<= beta 2.7e+152)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.7) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else if (beta <= 2.7e+152) {
		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 <= 7.7)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	elseif (beta <= 2.7e+152)
		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, 7.7], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.7e+152], 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 < 7.70000000000000018

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.6%

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

   if 7.70000000000000018 < beta < 2.70000000000000015e152

   1. Initial program 97.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 2.70000000000000015e152 < beta

   1. Initial program 74.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 87.8%

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

   10. Applied rewrites 88.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.3)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.3) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.3)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.3], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.29999999999999982

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.6%

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

   if 5.29999999999999982 < beta

   1. Initial program 84.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-+.f64 88.3

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

   8. Applied rewrites 88.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.7)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ 2.0 (+ alpha beta)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.7) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / (2.0 + (alpha + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.7)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(2.0 + Float64(alpha + beta)));
	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, 7.7], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.70000000000000018

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.6%

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

   if 7.70000000000000018 < beta

   1. Initial program 84.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

   7. Applied rewrites 88.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 93.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.7)
   (/ 0.16666666666666666 (+ 2.0 (+ alpha beta)))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.7) {
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	} 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 <= 7.7d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (alpha + beta))
    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 <= 7.7) {
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.7:
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta))
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.7)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(alpha + beta)));
	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 <= 7.7)
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	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, 7.7], N[(0.16666666666666666 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.70000000000000018

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.9%

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

   if 7.70000000000000018 < beta

   1. Initial program 84.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 91.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.7)
   (/ 0.16666666666666666 (+ 2.0 (+ alpha beta)))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.7) {
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	} 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 <= 7.7d0) then
        tmp = 0.16666666666666666d0 / (2.0d0 + (alpha + beta))
    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 <= 7.7) {
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.7:
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta))
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.7)
		tmp = Float64(0.16666666666666666 / Float64(2.0 + Float64(alpha + beta)));
	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 <= 7.7)
		tmp = 0.16666666666666666 / (2.0 + (alpha + beta));
	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, 7.7], N[(0.16666666666666666 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.70000000000000018

   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. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 66.9%

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

   if 7.70000000000000018 < beta

   1. Initial program 84.7%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 78.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.7:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.98:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.98) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.98) {
		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 <= 0.98d0) 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 <= 0.98) {
		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 <= 0.98:
		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 <= 0.98)
		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 <= 0.98)
		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, 0.98], 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 < 0.97999999999999998

   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 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 35.4%

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

   if 0.97999999999999998 < alpha

   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

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 19.8

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

   5. Applied rewrites 19.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 19.8%

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

Alternative 20: 32.0% 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 94.9%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
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 31.0

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

5. Applied rewrites 31.0%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 21.0%

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

Reproduce

herbie shell --seed 2024241 
(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)))