Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.1%

Time: 9.7s

Alternatives: 20

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {t\_1}^{-2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\left(\frac{\beta}{1 + \beta} - \frac{2}{-1 - \beta}\right) - \frac{1 + \beta}{{\left(-1 - \beta\right)}^{2}}}{\alpha} - \frac{-1}{1 + \beta}\right)}^{-1}}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= alpha 1.9e-13)
     (/ (* (+ (fma beta alpha (+ beta alpha)) 1.0) (pow t_1 -2.0)) t_0)
     (/
      (/
       (pow
        (-
         (/
          (-
           (- (/ beta (+ 1.0 beta)) (/ 2.0 (- -1.0 beta)))
           (/ (+ 1.0 beta) (pow (- -1.0 beta) 2.0)))
          alpha)
         (/ -1.0 (+ 1.0 beta)))
        -1.0)
       t_0)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (alpha <= 1.9e-13) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) * pow(t_1, -2.0)) / t_0;
	} else {
		tmp = (pow((((((beta / (1.0 + beta)) - (2.0 / (-1.0 - beta))) - ((1.0 + beta) / pow((-1.0 - beta), 2.0))) / alpha) - (-1.0 / (1.0 + beta))), -1.0) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (alpha <= 1.9e-13)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) * (t_1 ^ -2.0)) / t_0);
	else
		tmp = Float64(Float64((Float64(Float64(Float64(Float64(Float64(beta / Float64(1.0 + beta)) - Float64(2.0 / Float64(-1.0 - beta))) - Float64(Float64(1.0 + beta) / (Float64(-1.0 - beta) ^ 2.0))) / alpha) - Float64(-1.0 / Float64(1.0 + beta))) ^ -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[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.9e-13], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(N[(N[(beta / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + beta), $MachinePrecision] / N[Power[N[(-1.0 - beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(-1.0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.9e-13

   1. Initial program 99.9%

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

   4. Applied rewrites 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 1.9e-13 < alpha

   1. Initial program 87.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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 87.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 87.0

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lower-+.f64 87.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 87.0

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 87.0

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

      24. lift-fma.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-fma.f64 87.0

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

      27. lift-+.f64 N/A

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

      28. +-commutative N/A

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

      29. lift-+.f64 87.0

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

   6. Applied rewrites 87.0%

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

   7. Taylor expanded in alpha around -inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\left(\frac{\beta}{1 + \beta} - \frac{2}{-1 - \beta}\right) - \frac{1 + \beta}{{\left(-1 - \beta\right)}^{2}}}{\alpha} - \frac{-1}{1 + \beta}\right)}^{-1}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {t\_1}^{-2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.5e+137)
     (/ (* (+ (fma beta alpha (+ beta alpha)) 1.0) (pow t_1 -2.0)) t_0)
     (/
      (/
       (-
        (+ (+ 1.0 (+ alpha (pow beta -1.0))) (/ alpha beta))
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       t_0)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.5e+137) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) * pow(t_1, -2.0)) / t_0;
	} else {
		tmp = ((((1.0 + (alpha + pow(beta, -1.0))) + (alpha / beta)) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.5e+137)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) * (t_1 ^ -2.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(alpha + (beta ^ -1.0))) + Float64(alpha / beta)) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / t_0) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.5e+137], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[t$95$1, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + N[(alpha + N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e137

   1. Initial program 98.5%

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

   4. Applied rewrites 97.3%

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

   if 1.5e137 < beta

   1. Initial program 78.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 78.3%

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

   5. Taylor expanded in beta around inf

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

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1\right) \cdot {\left(\left(\beta + \alpha\right) + 2\right)}^{-2}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_1}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ beta alpha))) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.5e+137)
     (/ (/ (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_1) t_0) t_1)
     (/
      (/
       (-
        (+ (+ 1.0 (+ alpha (pow beta -1.0))) (/ alpha beta))
        (* (+ 1.0 alpha) (/ (+ 2.0 alpha) beta)))
       t_0)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (beta + alpha);
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.5e+137) {
		tmp = (((fma(beta, alpha, (beta + alpha)) + 1.0) / t_1) / t_0) / t_1;
	} else {
		tmp = ((((1.0 + (alpha + pow(beta, -1.0))) + (alpha / beta)) - ((1.0 + alpha) * ((2.0 + alpha) / beta))) / t_0) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.5e+137)
		tmp = Float64(Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_1) / t_0) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(alpha + (beta ^ -1.0))) + Float64(alpha / beta)) - Float64(Float64(1.0 + alpha) * Float64(Float64(2.0 + alpha) / beta))) / t_0) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.5e+137], N[(N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + N[(alpha + N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e137

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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 98.5%

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

   if 1.5e137 < beta

   1. Initial program 78.3%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 78.3%

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

   5. Taylor expanded in beta around inf

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

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \left(\alpha + {\beta}^{-1}\right)\right) + \frac{\alpha}{\beta}\right) - \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0}}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({\beta}^{-1} + 1\right) \cdot \alpha - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 6e+161)
     (/
      (/
       (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_0)
       (+ 3.0 (+ beta alpha)))
      t_0)
     (/
      (/
       (-
        (* (+ (pow beta -1.0) 1.0) alpha)
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      (+ (+ (+ alpha beta) 2.0) 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 6e+161) {
		tmp = (((fma(beta, alpha, (beta + alpha)) + 1.0) / t_0) / (3.0 + (beta + alpha))) / t_0;
	} else {
		tmp = ((((pow(beta, -1.0) + 1.0) * alpha) - ((1.0 + alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / (((alpha + beta) + 2.0) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 6e+161)
		tmp = Float64(Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_0) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64((beta ^ -1.0) + 1.0) * alpha) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / Float64(Float64(Float64(alpha + beta) + 2.0) + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 6e+161], N[(N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[beta, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] * alpha), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000023e161

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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 98.5%

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

   if 6.00000000000000023e161 < beta

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({\beta}^{-1} + 1\right) \cdot \alpha - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0}}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.5e+137)
     (/
      (/
       (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_0)
       (+ 3.0 (+ beta alpha)))
      t_0)
     (/ (/ (+ alpha 1.0) t_0) (* (- (/ (- -3.0 alpha) beta) 1.0) (- beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.5e+137) {
		tmp = (((fma(beta, alpha, (beta + alpha)) + 1.0) / t_0) / (3.0 + (beta + alpha))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / t_0) / ((((-3.0 - alpha) / beta) - 1.0) * -beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.5e+137)
		tmp = Float64(Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_0) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(Float64(Float64(Float64(-3.0 - alpha) / beta) - 1.0) * Float64(-beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.5e+137], N[(N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(N[(-3.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] - 1.0), $MachinePrecision] * (-beta)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e137

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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 98.5%

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

   if 1.5e137 < beta

   1. Initial program 78.3%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 90.3

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

   8. Applied rewrites 90.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

   10. Applied rewrites 90.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.5e+137)
     (/
      (/
       (+ 1.0 (fma alpha beta (+ alpha beta)))
       (* (+ 2.0 (+ alpha beta)) (+ 3.0 (+ alpha beta))))
      t_0)
     (/ (/ (+ alpha 1.0) t_0) (* (- (/ (- -3.0 alpha) beta) 1.0) (- beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.5e+137) {
		tmp = ((1.0 + fma(alpha, beta, (alpha + beta))) / ((2.0 + (alpha + beta)) * (3.0 + (alpha + beta)))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / t_0) / ((((-3.0 - alpha) / beta) - 1.0) * -beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.5e+137)
		tmp = Float64(Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / Float64(Float64(2.0 + Float64(alpha + beta)) * Float64(3.0 + Float64(alpha + beta)))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(Float64(Float64(Float64(-3.0 - alpha) / beta) - 1.0) * Float64(-beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.5e+137], N[(N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(N[(-3.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] - 1.0), $MachinePrecision] * (-beta)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e137

   1. Initial program 98.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 96.6%

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

   if 1.5e137 < beta

   1. Initial program 78.3%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 90.3

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

   8. Applied rewrites 90.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

   10. Applied rewrites 90.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2.15e+21)
     (/
      (+ (fma beta alpha (+ beta alpha)) 1.0)
      (* (* (+ 3.0 (+ beta alpha)) t_0) t_0))
     (/ (/ (+ alpha 1.0) t_0) (* (- (/ (- -3.0 alpha) beta) 1.0) (- beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2.15e+21) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha + 1.0) / t_0) / ((((-3.0 - alpha) / beta) - 1.0) * -beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2.15e+21)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(Float64(3.0 + Float64(beta + alpha)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(Float64(Float64(Float64(-3.0 - alpha) / beta) - 1.0) * Float64(-beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.15e+21], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(N[(-3.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] - 1.0), $MachinePrecision] * (-beta)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15e21

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.6%

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

   if 2.15e21 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 83.9

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

   8. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\alpha + \beta\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\alpha + 1\right)\right)}{\color{blue}{\left(\beta + \alpha\right)} + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)} \]

   10. Applied rewrites 83.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{-3 - \alpha}{\beta} - 1\right) \cdot \left(-\beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2.15e+21)
     (/
      (+ (fma beta alpha (+ beta alpha)) 1.0)
      (* (* (+ 3.0 (+ beta alpha)) t_0) t_0))
     (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2.15e+21) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2.15e+21)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(Float64(3.0 + Float64(beta + alpha)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.15e+21], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15e21

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.6%

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

   if 2.15e21 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+20)
   (/ (/ (+ 1.0 beta) (fma (+ 5.0 beta) beta 6.0)) (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+20) {
		tmp = ((1.0 + beta) / fma((5.0 + beta), beta, 6.0)) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+20)
		tmp = Float64(Float64(Float64(1.0 + beta) / fma(Float64(5.0 + beta), beta, 6.0)) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5e+20], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e20

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. 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 66.1

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

   7. Applied rewrites 66.1%

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

   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 66.1%

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

   if 1.5e20 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (/
    (fma
     (fma -0.05092592592592592 alpha 0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) (+ 3.0 (+ alpha beta))) (+ 2.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (alpha + beta))) / (2.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(alpha + beta))) / Float64(2.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.95], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.8%

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

   if 2.9500000000000002 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

   7. Applied rewrites 82.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{2 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (/
    (fma
     (fma -0.05092592592592592 alpha 0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ alpha beta))) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (alpha + beta))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(alpha + beta))) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.95], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.8%

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

   if 2.9500000000000002 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   7. Applied rewrites 82.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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 5.9)
     (/
      (fma
       (fma -0.05092592592592592 alpha 0.027777777777777776)
       alpha
       0.16666666666666666)
      t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5.9) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / t_0;
	} else {
		tmp = ((1.0 + alpha) / 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 <= 5.9)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5.9], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.9000000000000004

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.8%

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

   if 5.9000000000000004 < beta

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 84.8

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lower-+.f64 84.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 84.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 84.8

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

      24. lift-fma.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-fma.f64 84.8

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

      27. lift-+.f64 N/A

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

      28. +-commutative N/A

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

      29. lift-+.f64 84.8

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\beta + \alpha\right) + 2} \]
   8. 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 82.4

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

   9. Applied rewrites 82.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5)
   (/
    (fma
     (fma -0.05092592592592592 alpha 0.027777777777777776)
     alpha
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(fma(fma(-0.05092592592592592, alpha, 0.027777777777777776), alpha, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.5], N[(N[(N[(-0.05092592592592592 * alpha + 0.027777777777777776), $MachinePrecision] * alpha + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.8%

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

   if 7.5 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 82.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \alpha, 0.027777777777777776\right), \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.9)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (if (<= beta 4.6e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else if (beta <= 4.6e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.9)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 4.6e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.9], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.6e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 7.9000000000000004

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 64.7%

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

   if 7.9000000000000004 < beta < 4.6e154

   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 76.2

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

   5. Applied rewrites 76.2%

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

   if 4.6e154 < beta

   1. Initial program 76.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 88.7%

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

   10. Applied rewrites 86.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (if (<= beta 4.6e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 4.6e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else if (beta <= 4.6d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 4.6e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	elif beta <= 4.6e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 4.6e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	elseif (beta <= 4.6e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.6e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 65.2%

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

   if 8.5 < beta < 4.6e154

   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 76.2

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

   5. Applied rewrites 76.2%

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

   if 4.6e154 < beta

   1. Initial program 76.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 88.7%

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

   10. Applied rewrites 86.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.9)
   (/
    (fma 0.027777777777777776 alpha 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9) {
		tmp = fma(0.027777777777777776, alpha, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.9)
		tmp = Float64(fma(0.027777777777777776, alpha, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.9], N[(N[(0.027777777777777776 * alpha + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.9000000000000004

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 64.7%

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

   if 7.9000000000000004 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 82.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \alpha, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 93.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.5], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 65.2%

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

   if 8.5 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 90.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		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 <= 8.5d0) 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 <= 8.5) {
		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 <= 8.5:
		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 <= 8.5)
		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 <= 8.5)
		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, 8.5], 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 < 8.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 65.2%

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

   if 8.5 < beta

   1. Initial program 84.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 77.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.9e-13) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9e-13) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.9d-13) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9e-13) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.9e-13:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.9e-13)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.9e-13)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 1.9e-13], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.9e-13

   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 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 32.4%

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

   if 1.9e-13 < alpha

   1. Initial program 87.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 18.4%

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

Alternative 20: 32.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = alpha / (beta * beta)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 16.8%

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

Reproduce

herbie shell --seed 2024303 
(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)))