Octave 3.8, jcobi/3

Percentage Accurate: 94.9% → 99.8%

Time: 11.6s

Alternatives: 20

Speedup: 2.4×

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: 94.9% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 + \left(\alpha + \beta\right)\right)}^{-2} \cdot {\left(\frac{t\_0}{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(-1 - \left(\frac{1}{\beta} + \alpha\right)\right) - \frac{\alpha}{\beta}\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 4e+148)
     (*
      (pow (+ 2.0 (+ alpha beta)) -2.0)
      (pow (/ t_0 (- (fma beta alpha (+ alpha beta)) -1.0)) -1.0))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta))
        (- (- -1.0 (+ (/ 1.0 beta) alpha)) (/ alpha beta)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 4e+148) {
		tmp = pow((2.0 + (alpha + beta)), -2.0) * pow((t_0 / (fma(beta, alpha, (alpha + beta)) - -1.0)), -1.0);
	} else {
		tmp = ((((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta)) - ((-1.0 - ((1.0 / beta) + alpha)) - (alpha / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 4e+148)
		tmp = Float64((Float64(2.0 + Float64(alpha + beta)) ^ -2.0) * (Float64(t_0 / Float64(fma(beta, alpha, Float64(alpha + beta)) - -1.0)) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta)) - Float64(Float64(-1.0 - Float64(Float64(1.0 / beta) + alpha)) - Float64(alpha / beta))) / 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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 4e+148], N[(N[Power[N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(t$95$0 / N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000002e148

   1. Initial program 98.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-/r/ N/A

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

   4. Applied rewrites 98.5%

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

   if 4.0000000000000002e148 < beta

   1. Initial program 77.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 96.0

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

   5. Applied rewrites 96.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 96.0

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

   7. Applied rewrites 96.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 + \left(\alpha + \beta\right)\right)}^{-2} \cdot {\left(\frac{\left(\alpha + \beta\right) + 3}{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(-1 - \left(\frac{1}{\beta} + \alpha\right)\right) - \frac{\alpha}{\beta}\right)}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{t\_1}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(-1 - \left(\frac{1}{\beta} + \alpha\right)\right) - \frac{\alpha}{\beta}\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)) (t_1 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.8e+148)
     (/ (/ (- (fma beta alpha (+ alpha beta)) -1.0) t_1) (* t_1 t_0))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta))
        (- (- -1.0 (+ (/ 1.0 beta) alpha)) (/ alpha beta)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.8e+148) {
		tmp = ((fma(beta, alpha, (alpha + beta)) - -1.0) / t_1) / (t_1 * t_0);
	} else {
		tmp = ((((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta)) - ((-1.0 - ((1.0 / beta) + alpha)) - (alpha / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	t_1 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.8e+148)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(alpha + beta)) - -1.0) / t_1) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta)) - Float64(Float64(-1.0 - Float64(Float64(1.0 / beta) + alpha)) - Float64(alpha / beta))) / 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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.8e+148], N[(N[(N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998e148

   1. Initial program 98.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 97.8%

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

   if 3.7999999999999998e148 < beta

   1. Initial program 77.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 96.0

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

   5. Applied rewrites 96.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 96.0

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

   7. Applied rewrites 96.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(-1 - \left(\frac{1}{\beta} + \alpha\right)\right) - \frac{\alpha}{\beta}\right)}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{t\_0}}{t\_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{\left(\frac{\alpha - -3}{\beta} - -1\right) \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 3.8e+148)
     (/
      (/ (- (fma beta alpha (+ alpha beta)) -1.0) t_0)
      (* t_0 (+ (+ alpha beta) 3.0)))
     (/ (/ (- alpha -1.0) t_0) (* (- (/ (- alpha -3.0) beta) -1.0) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 3.8e+148) {
		tmp = ((fma(beta, alpha, (alpha + beta)) - -1.0) / t_0) / (t_0 * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((((alpha - -3.0) / beta) - -1.0) * beta);
	}
	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 <= 3.8e+148)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(alpha + beta)) - -1.0) / t_0) / Float64(t_0 * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0) * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.8e+148], N[(N[(N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998e148

   1. Initial program 98.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 97.8%

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

   if 3.7999999999999998e148 < beta

   1. Initial program 77.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in beta around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 96.3

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

   8. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 96.3

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

   10. Applied rewrites 96.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \left(\alpha + \beta\right)}}{\left(\frac{\alpha - -3}{\beta} - -1\right) \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{\left(t\_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{\left(\frac{\alpha - -3}{\beta} - -1\right) \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= beta 1e+18)
     (/
      (- (fma beta alpha (+ alpha beta)) -1.0)
      (* (* t_0 (+ (+ alpha beta) 3.0)) t_0))
     (/ (/ (- alpha -1.0) t_0) (* (- (/ (- alpha -3.0) beta) -1.0) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1e+18) {
		tmp = (fma(beta, alpha, (alpha + beta)) - -1.0) / ((t_0 * ((alpha + beta) + 3.0)) * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((((alpha - -3.0) / beta) - -1.0) * beta);
	}
	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 <= 1e+18)
		tmp = Float64(Float64(fma(beta, alpha, Float64(alpha + beta)) - -1.0) / Float64(Float64(t_0 * Float64(Float64(alpha + beta) + 3.0)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(Float64(Float64(Float64(alpha - -3.0) / beta) - -1.0) * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+18], N[(N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(N[(alpha - -3.0), $MachinePrecision] / beta), $MachinePrecision] - -1.0), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e18

   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 95.8%

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

   if 1e18 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in beta around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 89.7

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

   8. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 89.7

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

   10. Applied rewrites 89.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{\left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \left(\alpha + \beta\right)}}{\left(\frac{\alpha - -3}{\beta} - -1\right) \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{\left(t\_1 \cdot t\_0\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)) (t_1 (+ 2.0 (+ alpha beta))))
   (if (<= beta 1e+18)
     (/ (- (fma beta alpha (+ alpha beta)) -1.0) (* (* t_1 t_0) t_1))
     (/ (/ (- alpha -1.0) t_0) (+ (+ 2.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double t_1 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 1e+18) {
		tmp = (fma(beta, alpha, (alpha + beta)) - -1.0) / ((t_1 * t_0) * t_1);
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	t_1 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1e+18)
		tmp = Float64(Float64(fma(beta, alpha, Float64(alpha + beta)) - -1.0) / Float64(Float64(t_1 * t_0) * t_1));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+18], N[(N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e18

   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 95.8%

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

   if 1e18 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.7

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

   5. Applied rewrites 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 89.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) - -1}{\left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta - -1}{2 + \beta}}{2 + \left(\alpha + \beta\right)}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+15)
   (/ (/ (/ (- beta -1.0) (+ 2.0 beta)) (+ 2.0 (+ alpha beta))) (+ 3.0 beta))
   (/ (/ (- alpha -1.0) (+ (+ alpha beta) 3.0)) (+ (+ 2.0 beta) alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+15) {
		tmp = (((beta - -1.0) / (2.0 + beta)) / (2.0 + (alpha + beta))) / (3.0 + beta);
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.8d+15) then
        tmp = (((beta - (-1.0d0)) / (2.0d0 + beta)) / (2.0d0 + (alpha + beta))) / (3.0d0 + beta)
    else
        tmp = ((alpha - (-1.0d0)) / ((alpha + beta) + 3.0d0)) / ((2.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+15) {
		tmp = (((beta - -1.0) / (2.0 + beta)) / (2.0 + (alpha + beta))) / (3.0 + beta);
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.8e+15:
		tmp = (((beta - -1.0) / (2.0 + beta)) / (2.0 + (alpha + beta))) / (3.0 + beta)
	else:
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+15)
		tmp = Float64(Float64(Float64(Float64(beta - -1.0) / Float64(2.0 + beta)) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + beta));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 3.0)) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8e+15)
		tmp = (((beta - -1.0) / (2.0 + beta)) / (2.0 + (alpha + beta))) / (3.0 + beta);
	else
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8e+15], N[(N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e15

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.5

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

   7. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 89.5

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

   9. Applied rewrites 89.5%

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

   10. Taylor expanded in alpha around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 68.6

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

   12. Applied rewrites 68.6%

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

   if 3.8e15 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.7

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

   5. Applied rewrites 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 89.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta - -1}{2 + \beta}}{2 + \left(\alpha + \beta\right)}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{\beta - -1} \cdot \left(\left(3 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+34)
   (/ 1.0 (* (/ (+ 2.0 beta) (- beta -1.0)) (* (+ 3.0 beta) (+ 2.0 beta))))
   (/ (/ (- alpha -1.0) (+ (+ alpha beta) 3.0)) (+ (+ 2.0 beta) alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+34) {
		tmp = 1.0 / (((2.0 + beta) / (beta - -1.0)) * ((3.0 + beta) * (2.0 + beta)));
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5d+34) then
        tmp = 1.0d0 / (((2.0d0 + beta) / (beta - (-1.0d0))) * ((3.0d0 + beta) * (2.0d0 + beta)))
    else
        tmp = ((alpha - (-1.0d0)) / ((alpha + beta) + 3.0d0)) / ((2.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+34) {
		tmp = 1.0 / (((2.0 + beta) / (beta - -1.0)) * ((3.0 + beta) * (2.0 + beta)));
	} else {
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5e+34:
		tmp = 1.0 / (((2.0 + beta) / (beta - -1.0)) * ((3.0 + beta) * (2.0 + beta)))
	else:
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+34)
		tmp = Float64(1.0 / Float64(Float64(Float64(2.0 + beta) / Float64(beta - -1.0)) * Float64(Float64(3.0 + beta) * Float64(2.0 + beta))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 3.0)) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e+34)
		tmp = 1.0 / (((2.0 + beta) / (beta - -1.0)) * ((3.0 + beta) * (2.0 + beta)));
	else
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5e+34], N[(1.0 / N[(N[(N[(2.0 + beta), $MachinePrecision] / N[(beta - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999998e34

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if 4.9999999999999998e34 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 89.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\frac{2 + \beta}{\beta - -1} \cdot \left(\left(3 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 13.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(2 + \alpha\right) + \beta}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 13.5)
     (/ (/ (fma 0.25 beta 0.5) (+ (+ 2.0 alpha) beta)) t_0)
     (/ (/ (- alpha -1.0) t_0) (+ (+ 2.0 beta) alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 13.5) {
		tmp = (fma(0.25, beta, 0.5) / ((2.0 + alpha) + beta)) / t_0;
	} else {
		tmp = ((alpha - -1.0) / t_0) / ((2.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 13.5)
		tmp = Float64(Float64(fma(0.25, beta, 0.5) / Float64(Float64(2.0 + alpha) + beta)) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 13.5], N[(N[(N[(0.25 * beta + 0.5), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 13.5

   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

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.2

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

   7. Applied rewrites 89.2%

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

   9. Applied rewrites 89.2%

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

   10. Taylor expanded in beta around 0

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

   12. Applied rewrites 88.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right)}{\left(2 + \alpha\right) + \beta}}{3 + \left(\beta + \alpha\right)} \]

   if 13.5 < beta

   1. Initial program 85.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 88.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 13.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25, \beta, 0.5\right)}{\left(2 + \alpha\right) + \beta}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.0d0) then
        tmp = ((alpha - (-1.0d0)) / (2.0d0 + alpha)) / ((alpha + 3.0d0) * (2.0d0 + alpha))
    else
        tmp = ((alpha - (-1.0d0)) / ((alpha + beta) + 3.0d0)) / ((2.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.0:
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((alpha + 3.0) * (2.0 + alpha))
	else:
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.0)
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(2.0 + alpha)) / Float64(Float64(alpha + 3.0) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(alpha + beta) + 3.0)) / Float64(Float64(2.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.0)
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((alpha + 3.0) * (2.0 + alpha));
	else
		tmp = ((alpha - -1.0) / ((alpha + beta) + 3.0)) / ((2.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.0], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 3.0), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $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. 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 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in beta around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-+.f64 98.2

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

   13. Applied rewrites 98.2%

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

   if 5 < beta

   1. Initial program 85.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 88.8%

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

4. Final simplification 95.0%

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

Alternative 10: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \beta\right) + \alpha\\ t_1 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 beta) alpha)) (t_1 (+ (+ alpha beta) 3.0)))
   (if (<= beta 15.2)
     (/ (fma 0.25 alpha 0.5) (* t_0 t_1))
     (/ (/ (- alpha -1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + beta) + alpha;
	double t_1 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 15.2) {
		tmp = fma(0.25, alpha, 0.5) / (t_0 * t_1);
	} else {
		tmp = ((alpha - -1.0) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + beta) + alpha)
	t_1 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 15.2)
		tmp = Float64(fma(0.25, alpha, 0.5) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 15.2], N[(N[(0.25 * alpha + 0.5), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 15.199999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 86.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\alpha}, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 15.199999999999999 < beta

   1. Initial program 85.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 88.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\alpha + \beta\right) + 3}}{\left(2 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 15.2)
     (/ (fma 0.25 alpha 0.5) (* (+ (+ 2.0 beta) alpha) t_0))
     (/ (/ (- alpha -1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 15.2) {
		tmp = fma(0.25, alpha, 0.5) / (((2.0 + beta) + alpha) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 15.2)
		tmp = Float64(fma(0.25, alpha, 0.5) / Float64(Float64(Float64(2.0 + beta) + alpha) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / 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[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 15.2], N[(N[(0.25 * alpha + 0.5), $MachinePrecision] / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 15.199999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 86.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\alpha}, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 15.199999999999999 < beta

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 85.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 85.1

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.1

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

   7. Applied rewrites 89.1%

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

   9. Applied rewrites 89.1%

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

   10. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 88.4

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

   12. Applied rewrites 88.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 3\\ \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 3.0)))
   (if (<= beta 15.2)
     (/ 0.5 (* (+ (+ 2.0 beta) alpha) t_0))
     (/ (/ (- alpha -1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 15.2) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 3.0d0
    if (beta <= 15.2d0) then
        tmp = 0.5d0 / (((2.0d0 + beta) + alpha) * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 3.0;
	double tmp;
	if (beta <= 15.2) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 3.0
	tmp = 0
	if beta <= 15.2:
		tmp = 0.5 / (((2.0 + beta) + alpha) * t_0)
	else:
		tmp = ((alpha - -1.0) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 3.0)
	tmp = 0.0
	if (beta <= 15.2)
		tmp = Float64(0.5 / Float64(Float64(Float64(2.0 + beta) + alpha) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 3.0;
	tmp = 0.0;
	if (beta <= 15.2)
		tmp = 0.5 / (((2.0 + beta) + alpha) * t_0);
	else
		tmp = ((alpha - -1.0) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[beta, 15.2], N[(0.5 / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 15.199999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 87.8%

       \[\leadsto \frac{0.5}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 15.199999999999999 < beta

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 85.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 85.1

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.1

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

   7. Applied rewrites 89.1%

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

   9. Applied rewrites 89.1%

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

   10. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 88.4

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

   12. Applied rewrites 88.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 15.2:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 16:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 16.0)
   (/ 0.5 (* (+ (+ 2.0 beta) alpha) (+ (+ alpha beta) 3.0)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 16.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 16.0d0) then
        tmp = 0.5d0 / (((2.0d0 + beta) + alpha) * ((alpha + beta) + 3.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 16.0) {
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 16.0:
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((alpha + beta) + 3.0))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 16.0)
		tmp = Float64(0.5 / Float64(Float64(Float64(2.0 + beta) + alpha) * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 16.0)
		tmp = 0.5 / (((2.0 + beta) + alpha) * ((alpha + beta) + 3.0));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 16.0], N[(0.5 / N[(N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 16

   1. Initial program 99.9%

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

   3. 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 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 87.8%

       \[\leadsto \frac{0.5}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \beta\right) + \alpha\right)} \]

   if 16 < beta

   1. Initial program 85.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 88.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 16:\\ \;\;\;\;\frac{0.5}{\left(\left(2 + \beta\right) + \alpha\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+148}:\\ \;\;\;\;\frac{\alpha - -1}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+148)
   (/ (- alpha -1.0) (* (+ 3.0 beta) (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+148) {
		tmp = (alpha - -1.0) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3d+148) then
        tmp = (alpha - (-1.0d0)) / ((3.0d0 + beta) * (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+148) {
		tmp = (alpha - -1.0) / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3e+148:
		tmp = (alpha - -1.0) / ((3.0 + beta) * (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+148)
		tmp = Float64(Float64(alpha - -1.0) / Float64(Float64(3.0 + beta) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+148)
		tmp = (alpha - -1.0) / ((3.0 + beta) * (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3e+148], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.00000000000000015e148

   1. Initial program 98.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 25.3

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

   10. Applied rewrites 25.3%

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

   if 3.00000000000000015e148 < beta

   1. Initial program 77.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+148}:\\ \;\;\;\;\frac{\alpha - -1}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.5e-14) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.5e-14) {
		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 <= 8.5d-14) 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 <= 8.5e-14) {
		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 <= 8.5e-14:
		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 <= 8.5e-14)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8.5e-14)
		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, 8.5e-14], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.50000000000000038e-14

   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 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 37.3%

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

   10. Applied rewrites 37.5%

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

   if 8.50000000000000038e-14 < alpha

   1. Initial program 84.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 20.0%

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

   10. Applied rewrites 21.2%

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

Alternative 16: 54.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d+154) then
        tmp = (alpha - (-1.0d0)) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35e+154) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.35e+154:
		tmp = (alpha - -1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35e+154)
		tmp = (alpha - -1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1.35000000000000003e154

   1. Initial program 98.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 17.6

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

   5. Applied rewrites 17.6%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 77.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 95.0%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\frac{\alpha - -1}{\beta}}{\beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha - -1.0) / 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 - (-1.0d0)) / beta) / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((alpha - -1.0) / beta) / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((alpha - -1.0) / beta) / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(alpha - -1.0) / beta) / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha - -1.0) / beta) / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $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 31.7

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

5. Applied rewrites 31.7%

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

7. Applied rewrites 32.3%

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

8. Final simplification 32.3%

   \[\leadsto \frac{\frac{\alpha - -1}{\beta}}{\beta} \]
9. Add Preprocessing

Alternative 18: 51.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.5e-14) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.5e-14) {
		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 <= 8.5d-14) 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 <= 8.5e-14) {
		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 <= 8.5e-14:
		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 <= 8.5e-14)
		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 <= 8.5e-14)
		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, 8.5e-14], 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 < 8.50000000000000038e-14

   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 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 37.3%

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

   if 8.50000000000000038e-14 < alpha

   1. Initial program 84.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 20.0%

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

Alternative 19: 52.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\alpha - -1}{\beta \cdot \beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (alpha - -1.0) / (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 - (-1.0d0)) / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (alpha - -1.0) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (alpha - -1.0) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(alpha - -1.0) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (alpha - -1.0) / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(alpha - -1.0), $MachinePrecision] / 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 31.7

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

5. Applied rewrites 31.7%

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

6. Final simplification 31.7%

   \[\leadsto \frac{\alpha - -1}{\beta \cdot \beta} \]
7. Add Preprocessing

Alternative 20: 31.5% 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 31.7

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

5. Applied rewrites 31.7%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 21.3%

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

Reproduce

herbie shell --seed 2024255 
(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)))