Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 99.5%

Time: 10.8s

Alternatives: 24

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \left(\beta + \alpha\right) + 3\\ \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;{t\_0}^{-2} \cdot {\left(\frac{t\_1}{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ (+ beta alpha) 3.0)))
   (if (<= alpha 4.3e+32)
     (*
      (pow t_0 -2.0)
      (pow (/ t_1 (- (fma beta alpha (+ beta alpha)) -1.0)) -1.0))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (+ 2.0 alpha) beta))
        (- (- (/ -1.0 beta) (/ alpha beta)) (- alpha -1.0)))
       t_1)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (beta + alpha) + 3.0;
	double tmp;
	if (alpha <= 4.3e+32) {
		tmp = pow(t_0, -2.0) * pow((t_1 / (fma(beta, alpha, (beta + alpha)) - -1.0)), -1.0);
	} else {
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(Float64(beta + alpha) + 3.0)
	tmp = 0.0
	if (alpha <= 4.3e+32)
		tmp = Float64((t_0 ^ -2.0) * (Float64(t_1 / Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0)) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(Float64(2.0 + alpha) / beta)) - Float64(Float64(Float64(-1.0 / beta) - Float64(alpha / beta)) - 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[alpha, 4.3e+32], N[(N[Power[t$95$0, -2.0], $MachinePrecision] * N[Power[N[(t$95$1 / N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 / beta), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2999999999999997e32

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. 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 99.9%

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

   if 4.2999999999999997e32 < alpha

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 24.4

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

   7. Applied rewrites 24.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;{\left(\left(\beta + \alpha\right) + 2\right)}^{-2} \cdot {\left(\frac{\left(\beta + \alpha\right) + 3}{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{t\_0}}{t\_0}}{t\_0 - -1}\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1
         (/
          (/ (/ (- (+ (* beta alpha) (+ beta alpha)) -1.0) t_0) t_0)
          (- t_0 -1.0))))
   (if (<= t_1 0.1)
     t_1
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 5.0) beta))
        (- (- (/ -1.0 beta) (/ alpha beta)) (- alpha -1.0)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = ((((-1.0 - alpha) * (fma(2.0, alpha, 5.0) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / beta) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(beta * alpha) + Float64(beta + alpha)) - -1.0) / t_0) / t_0) / Float64(t_0 - -1.0))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 5.0) / beta)) - Float64(Float64(Float64(-1.0 / beta) - Float64(alpha / beta)) - 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(beta * alpha), $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]}, If[LessEqual[t$95$1, 0.1], t$95$1, N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 5.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 / beta), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 1.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 1.7%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 57.6

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

   7. Applied rewrites 57.6%

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

4. Final simplification 97.2%

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

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 3\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{{t\_1}^{-2} \cdot \left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 3.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= alpha 4.3e+32)
     (/ (* (pow t_1 -2.0) (- (fma beta alpha (+ beta alpha)) -1.0)) t_0)
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (+ 2.0 alpha) beta))
        (- (- (/ -1.0 beta) (/ alpha beta)) (- alpha -1.0)))
       t_0)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 3.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (alpha <= 4.3e+32) {
		tmp = (pow(t_1, -2.0) * (fma(beta, alpha, (beta + alpha)) - -1.0)) / t_0;
	} else {
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 3.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (alpha <= 4.3e+32)
		tmp = Float64(Float64((t_1 ^ -2.0) * Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(Float64(2.0 + alpha) / beta)) - Float64(Float64(Float64(-1.0 / beta) - Float64(alpha / beta)) - Float64(alpha - -1.0))) / t_0) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[alpha, 4.3e+32], N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 / beta), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2999999999999997e32

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   if 4.2999999999999997e32 < alpha

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 24.4

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

   7. Applied rewrites 24.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{{\left(\left(\beta + \alpha\right) + 2\right)}^{-2} \cdot \left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1\right)}{\left(\beta + \alpha\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{t\_0}}{t\_0}}{t\_0 - -1}\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1
         (/
          (/ (/ (- (+ (* beta alpha) (+ beta alpha)) -1.0) t_0) t_0)
          (- t_0 -1.0))))
   (if (<= t_1 0.1) t_1 (/ (/ (- alpha -1.0) (+ (+ beta alpha) 3.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = (((((beta * alpha) + (beta + alpha)) - (-1.0d0)) / t_0) / t_0) / (t_0 - (-1.0d0))
    if (t_1 <= 0.1d0) then
        tmp = t_1
    else
        tmp = ((alpha - (-1.0d0)) / ((beta + alpha) + 3.0d0)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0)
	tmp = 0
	if t_1 <= 0.1:
		tmp = t_1
	else:
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(beta * alpha) + Float64(beta + alpha)) - -1.0) / t_0) / t_0) / Float64(t_0 - -1.0))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(beta + alpha) + 3.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(beta * alpha), $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]}, If[LessEqual[t$95$1, 0.1], t$95$1, N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 1.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 1.7%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 58.3

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

   7. Applied rewrites 58.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{t\_0}}{t\_0}}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{\left(\beta + \alpha\right) + 3}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= alpha 4.3e+32)
     (/
      (/ (/ (- (+ (* beta alpha) (+ beta alpha)) -1.0) t_0) t_0)
      (- t_0 -1.0))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (+ 2.0 alpha) beta))
        (- (- (/ -1.0 beta) (/ alpha beta)) (- alpha -1.0)))
       (+ (+ beta alpha) 3.0))
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (alpha <= 4.3e+32) {
		tmp = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	} else {
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (alpha <= 4.3d+32) then
        tmp = (((((beta * alpha) + (beta + alpha)) - (-1.0d0)) / t_0) / t_0) / (t_0 - (-1.0d0))
    else
        tmp = (((((-1.0d0) - alpha) * ((2.0d0 + alpha) / beta)) - ((((-1.0d0) / beta) - (alpha / beta)) - (alpha - (-1.0d0)))) / ((beta + alpha) + 3.0d0)) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (alpha <= 4.3e+32) {
		tmp = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	} else {
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if alpha <= 4.3e+32:
		tmp = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0)
	else:
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / ((beta + alpha) + 3.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (alpha <= 4.3e+32)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta * alpha) + Float64(beta + alpha)) - -1.0) / t_0) / t_0) / Float64(t_0 - -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(Float64(2.0 + alpha) / beta)) - Float64(Float64(Float64(-1.0 / beta) - Float64(alpha / beta)) - Float64(alpha - -1.0))) / Float64(Float64(beta + alpha) + 3.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (alpha <= 4.3e+32)
		tmp = (((((beta * alpha) + (beta + alpha)) - -1.0) / t_0) / t_0) / (t_0 - -1.0);
	else
		tmp = ((((-1.0 - alpha) * ((2.0 + alpha) / beta)) - (((-1.0 / beta) - (alpha / beta)) - (alpha - -1.0))) / ((beta + alpha) + 3.0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[alpha, 4.3e+32], N[(N[(N[(N[(N[(N[(beta * alpha), $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], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 / beta), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2999999999999997e32

   1. Initial program 99.8%

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

   if 4.2999999999999997e32 < alpha

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 24.4

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

   7. Applied rewrites 24.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) - -1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{2 + \alpha}{\beta} - \left(\left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right) - \left(\alpha - -1\right)\right)}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\mathsf{fma}\left(\mathsf{fma}\left(2, \alpha, \beta\right) + 5, \beta, \left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+30)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (*
       (fma (+ (fma 2.0 alpha beta) 5.0) beta (* (+ 3.0 alpha) (+ 2.0 alpha)))
       t_0))
     (/ (/ (- alpha -1.0) (+ (+ beta alpha) 3.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+30) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (fma((fma(2.0, alpha, beta) + 5.0), beta, ((3.0 + alpha) * (2.0 + alpha))) * t_0);
	} else {
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+30)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(fma(Float64(fma(2.0, alpha, beta) + 5.0), beta, Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha))) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(beta + alpha) + 3.0)) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+30], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(N[(2.0 * alpha + beta), $MachinePrecision] + 5.0), $MachinePrecision] * beta + N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999998e30

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. 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 95.9%

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

   5. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 96.0

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

   7. Applied rewrites 96.0%

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

   if 4.9999999999999998e30 < beta

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\mathsf{fma}\left(\mathsf{fma}\left(2, \alpha, \beta\right) + 5, \beta, \left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\mathsf{fma}\left(t\_0, t\_0, t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+30)
     (/ (- (fma beta alpha (+ beta alpha)) -1.0) (* (fma t_0 t_0 t_0) t_0))
     (/ (/ (- alpha -1.0) (+ (+ beta alpha) 3.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+30) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (fma(t_0, t_0, t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / ((beta + alpha) + 3.0)) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+30)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(fma(t_0, t_0, t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(beta + alpha) + 3.0)) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+30], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(t$95$0 * t$95$0 + t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999998e30

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. 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 95.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 96.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.0

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 96.0

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

      26. lift-+.f64 N/A

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

      27. +-commutative N/A

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

      28. lift-+.f64 96.0

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

   6. Applied rewrites 96.0%

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

   if 4.9999999999999998e30 < beta

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\mathsf{fma}\left(\left(\beta + \alpha\right) + 2, \left(\beta + \alpha\right) + 2, \left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 3\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_1}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 3.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 4e+102)
     (/ (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_1) (* t_1 t_0))
     (/ (/ (- alpha -1.0) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 3.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4e+102) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_1) / (t_1 * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 3.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 4e+102)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_1) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+102], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999991e102

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.0%

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

   if 3.99999999999999991e102 < beta

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 91.9

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

   7. Applied rewrites 91.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 3\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(t\_1 \cdot t\_0\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 3.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+30)
     (/ (- (fma beta alpha (+ beta alpha)) -1.0) (* (* t_1 t_0) t_1))
     (/ (/ (- alpha -1.0) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 3.0;
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+30) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / ((t_1 * t_0) * t_1);
	} else {
		tmp = ((alpha - -1.0) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 3.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+30)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(Float64(t_1 * t_0) * t_1));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+30], N[(N[(N[(beta * alpha + N[(beta + alpha), $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] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999998e30

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. 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 95.9%

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

   if 4.9999999999999998e30 < beta

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 3}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(2 + \beta\right) + \alpha}}{\left(3 + \alpha\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2e+16)
   (/ (/ (- beta -1.0) (fma (+ 5.0 beta) beta 6.0)) (+ (+ beta alpha) 2.0))
   (/ (/ (- alpha -1.0) (+ (+ 2.0 beta) alpha)) (+ (+ 3.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2e+16) {
		tmp = ((beta - -1.0) / fma((5.0 + beta), beta, 6.0)) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2e+16)
		tmp = Float64(Float64(Float64(beta - -1.0) / fma(Float64(5.0 + beta), beta, 6.0)) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(2.0 + beta) + alpha)) / Float64(Float64(3.0 + alpha) + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2e+16], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 68.7

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 68.7%

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

   if 5.2e16 < beta

   1. Initial program 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.6

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

   7. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 86.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(2 + \beta\right) + \alpha}}{\left(3 + \alpha\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+16) {
		tmp = ((beta - -1.0) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	} else {
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + 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 <= 1.15d+16) then
        tmp = ((beta - (-1.0d0)) / ((beta + 3.0d0) * (2.0d0 + beta))) / (2.0d0 + beta)
    else
        tmp = ((alpha - (-1.0d0)) / ((2.0d0 + beta) + alpha)) / ((3.0d0 + alpha) + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+16) {
		tmp = ((beta - -1.0) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	} else {
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15e+16:
		tmp = ((beta - -1.0) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta)
	else:
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15e+16)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(Float64(beta + 3.0) * Float64(2.0 + beta))) / Float64(2.0 + beta));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(2.0 + beta) + alpha)) / Float64(Float64(3.0 + alpha) + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15e+16)
		tmp = ((beta - -1.0) / ((beta + 3.0) * (2.0 + beta))) / (2.0 + beta);
	else
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + 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, 1.15e+16], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 68.5

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

   7. Applied rewrites 68.5%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 67.7

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

   10. Applied rewrites 67.7%

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

   if 1.15e16 < beta

   1. Initial program 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.8

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

   7. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 86.8%

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

4. Final simplification 74.1%

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

Alternative 12: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(2 + \beta\right) + \alpha}}{\left(3 + \alpha\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (/
    (fma
     (fma
      (fma 0.03780864197530864 beta -0.05092592592592592)
      beta
      0.027777777777777776)
     beta
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (- alpha -1.0) (+ (+ 2.0 beta) alpha)) (+ (+ 3.0 alpha) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((alpha - -1.0) / ((2.0 + beta) + alpha)) / ((3.0 + alpha) + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(Float64(2.0 + beta) + alpha)) / Float64(Float64(3.0 + alpha) + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(N[(N[(N[(0.03780864197530864 * beta + -0.05092592592592592), $MachinePrecision] * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.7

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

   10. Applied rewrites 67.7%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 67.5%

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

   if 1.5 < beta

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

   7. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 83.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(2 + \beta\right) + \alpha}}{\left(3 + \alpha\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2.5)
     (/
      (fma
       (fma
        (fma 0.03780864197530864 beta -0.05092592592592592)
        beta
        0.027777777777777776)
       beta
       0.16666666666666666)
      t_0)
     (/ (/ (- alpha -1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2.5) {
		tmp = fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / 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(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.5], N[(N[(N[(N[(0.03780864197530864 * beta + -0.05092592592592592), $MachinePrecision] * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.7

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

   10. Applied rewrites 67.7%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 67.5%

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

   if 2.5 < beta

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 83.3

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

   7. Applied rewrites 83.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/
    (fma 0.027777777777777776 beta 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (if (<= beta 6.3e+152)
     (/ (- alpha -1.0) (* beta beta))
     (if (<= beta 6.5e+161) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = fma(0.027777777777777776, beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else if (beta <= 6.3e+152) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else if (beta <= 6.5e+161) {
		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 (beta <= 5.4)
		tmp = Float64(fma(0.027777777777777776, beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 6.3e+152)
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	elseif (beta <= 6.5e+161)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(N[(0.027777777777777776 * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.3e+152], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.5e+161], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.3

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 66.4%

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

   if 5.4000000000000004 < beta < 6.29999999999999957e152

   1. Initial program 91.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 6.29999999999999957e152 < beta < 6.5e161

   1. Initial program 99.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 48.5%

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

   10. Applied rewrites 99.1%

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

   if 6.5e161 < beta

   1. Initial program 68.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 91.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (if (<= beta 6.3e+152)
     (/ (- alpha -1.0) (* beta beta))
     (if (<= beta 6.5e+161) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 6.3e+152) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else if (beta <= 6.5e+161) {
		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 (beta <= 9.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else if (beta <= 6.3d+152) then
        tmp = (alpha - (-1.0d0)) / (beta * beta)
    else if (beta <= 6.5d+161) 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 (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 6.3e+152) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else if (beta <= 6.5e+161) {
		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 beta <= 9.0:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	elif beta <= 6.3e+152:
		tmp = (alpha - -1.0) / (beta * beta)
	elif beta <= 6.5e+161:
		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 (beta <= 9.0)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 6.3e+152)
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	elseif (beta <= 6.5e+161)
		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 (beta <= 9.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	elseif (beta <= 6.3e+152)
		tmp = (alpha - -1.0) / (beta * beta);
	elseif (beta <= 6.5e+161)
		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[beta, 9.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.3e+152], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.5e+161], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 9

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.3

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 65.6%

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

   if 9 < beta < 6.29999999999999957e152

   1. Initial program 91.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 6.29999999999999957e152 < beta < 6.5e161

   1. Initial program 99.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 48.5%

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

   10. Applied rewrites 99.1%

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

   if 6.5e161 < beta

   1. Initial program 68.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 91.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.85)
     (/
      (fma
       (fma -0.05092592592592592 beta 0.027777777777777776)
       beta
       0.16666666666666666)
      t_0)
     (/ (/ (- alpha -1.0) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.85) {
		tmp = fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 0.16666666666666666) / 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(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.85)
		tmp = Float64(fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 0.16666666666666666) / 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.85], N[(N[(N[(-0.05092592592592592 * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8500000000000001

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.7

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

   10. Applied rewrites 67.7%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 67.3%

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

   if 1.8500000000000001 < beta

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 83.3

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

   7. Applied rewrites 83.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{2 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.9)
   (/
    (fma
     (fma -0.05092592592592592 beta 0.027777777777777776)
     beta
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (- alpha -1.0) beta) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.9) {
		tmp = fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (2.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.9)
		tmp = Float64(fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(2.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.9], N[(N[(N[(-0.05092592592592592 * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8999999999999999

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.7

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

   10. Applied rewrites 67.7%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 67.3%

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

   if 1.8999999999999999 < beta

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 83.2

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

   10. Applied rewrites 83.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{2 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (if (<= beta 3.4e+157)
     (/ (- alpha -1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 3.4e+157) {
		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 <= 9.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else if (beta <= 3.4d+157) 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 <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 3.4e+157) {
		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 <= 9.0:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	elif beta <= 3.4e+157:
		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 <= 9.0)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 3.4e+157)
		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 <= 9.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	elseif (beta <= 3.4e+157)
		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, 9.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.4e+157], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.3

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 65.6%

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

   if 9 < beta < 3.39999999999999979e157

   1. Initial program 92.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   if 3.39999999999999979e157 < beta

   1. Initial program 68.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 91.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 97.0% accurate, 2.4× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0)
   (/
    (fma 0.027777777777777776 beta 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (- alpha -1.0) beta) (+ 2.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = fma(0.027777777777777776, beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (2.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(fma(0.027777777777777776, beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(2.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(N[(0.027777777777777776 * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.3

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 66.4%

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

   if 4 < beta

   1. Initial program 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 84.2

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 84.0

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

   10. Applied rewrites 84.0%

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

4. Final simplification 72.7%

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

Alternative 20: 96.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.4)
   (/
    (fma 0.027777777777777776 beta 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.4) {
		tmp = fma(0.027777777777777776, beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.4)
		tmp = Float64(fma(0.027777777777777776, beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.4], N[(N[(0.027777777777777776 * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4000000000000004

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.3

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in beta around 0

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

   13. Applied rewrites 66.4%

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

   if 5.4000000000000004 < beta

   1. Initial program 82.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 84.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 93.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (/ (- alpha -1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = (alpha - -1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else
        tmp = (alpha - (-1.0d0)) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = (alpha - -1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	else:
		tmp = (alpha - -1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	else
		tmp = (alpha - -1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\mathsf{neg}\left(\left(\alpha + 1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{-\left(\mathsf{neg}\left(\color{blue}{\left(1 + \alpha\right)}\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      7. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      9. unsub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      10. lower--.f64 14.7

          \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 14.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      6. lower-+.f64 67.3

         \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

   10. Applied rewrites 67.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{6}}{\left(\beta + \alpha\right) + 2} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.6%

       \[\leadsto \frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2} \]

   if 9 < beta

   1. Initial program 82.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 75.0

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 91.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}} \]

   5. Taylor expanded in beta around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \alpha - 1\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\left(-1 \cdot \alpha - 1\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\mathsf{neg}\left(\left(\alpha + 1\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{-\left(\mathsf{neg}\left(\color{blue}{\left(1 + \alpha\right)}\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      7. distribute-neg-in N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{-\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\alpha\right)\right)\right)}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      9. unsub-neg N/A

         \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

      10. lower--.f64 14.7

          \[\leadsto \frac{\frac{-\color{blue}{\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   7. Applied rewrites 14.7%

      \[\leadsto \frac{\frac{\color{blue}{-\left(-1 - \alpha\right)}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}}}{\left(\beta + \alpha\right) + 2} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \beta}{2 + \beta}}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \beta}}{2 + \beta}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{1 + \beta}{\color{blue}{2 + \beta}}}{3 + \beta}}{\left(\beta + \alpha\right) + 2} \]

      6. lower-+.f64 67.3

         \[\leadsto \frac{\frac{\frac{1 + \beta}{2 + \beta}}{\color{blue}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

   10. Applied rewrites 67.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1 + \beta}{2 + \beta}}{3 + \beta}}}{\left(\beta + \alpha\right) + 2} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{6}}{\left(\beta + \alpha\right) + 2} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.6%

       \[\leadsto \frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2} \]

   if 9 < beta

   1. Initial program 82.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 75.0

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.3%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 52.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.8e-15) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.8e-15) {
		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 <= 7.8d-15) 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 <= 7.8e-15) {
		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 <= 7.8e-15:
		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 <= 7.8e-15)
		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 <= 7.8e-15)
		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, 7.8e-15], 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 < 7.80000000000000053e-15

   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 35.2

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.2%

      \[\leadsto \frac{1}{\color{blue}{\beta} \cdot \beta} \]

   if 7.80000000000000053e-15 < alpha

   1. Initial program 82.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 18.1

         \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.4%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 32.1% 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 93.7%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \alpha}}{{\beta}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

   4. lower-*.f64 29.3

      \[\leadsto \frac{1 + \alpha}{\color{blue}{\beta \cdot \beta}} \]

5. Applied rewrites 29.3%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

6. Taylor expanded in alpha around inf

   \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 17.1%

   \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))