Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.6%

Time: 10.4s

Alternatives: 16

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 13000000000000:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} + {\beta}^{-1}\right)\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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 13000000000000.0)
     (/
      (* (+ 1.0 beta) (+ 1.0 alpha))
      (*
       (fma (+ (+ alpha beta) 3.0) (+ alpha beta) (fma (+ alpha beta) 2.0 6.0))
       t_0))
     (/
      (/
       (-
        (+ (+ 1.0 alpha) (+ (/ alpha beta) (pow beta -1.0)))
        (* (+ 1.0 alpha) (/ (fma 2.0 alpha 5.0) beta)))
       beta)
      t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 13000000000000.0) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (fma(((alpha + beta) + 3.0), (alpha + beta), fma((alpha + beta), 2.0, 6.0)) * t_0);
	} else {
		tmp = ((((1.0 + alpha) + ((alpha / beta) + pow(beta, -1.0))) - ((1.0 + alpha) * (fma(2.0, alpha, 5.0) / beta))) / 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 <= 13000000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(fma(Float64(Float64(alpha + beta) + 3.0), Float64(alpha + beta), fma(Float64(alpha + beta), 2.0, 6.0)) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + alpha) + Float64(Float64(alpha / beta) + (beta ^ -1.0))) - Float64(Float64(1.0 + alpha) * Float64(fma(2.0, alpha, 5.0) / beta))) / beta) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 13000000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 2.0 + 6.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(N[(alpha / beta), $MachinePrecision] + N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 5.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3e13

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in beta around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 94.9

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

   7. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 N/A

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

      23. metadata-eval 94.9

          \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, \color{blue}{6}\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   9. Applied rewrites 94.9%

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right)} \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   if 1.3e13 < beta

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 13000000000000:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(1 + \alpha\right) + \left(\frac{\alpha}{\beta} + {\beta}^{-1}\right)\right) - \left(1 + \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 5\right)}{\beta}}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.52e+16)
     (/
      (* (+ 1.0 beta) (+ 1.0 alpha))
      (*
       (fma (+ (+ alpha beta) 3.0) (+ alpha beta) (fma (+ alpha beta) 2.0 6.0))
       t_0))
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.52e+16) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (fma(((alpha + beta) + 3.0), (alpha + beta), fma((alpha + beta), 2.0, 6.0)) * t_0);
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.52e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(fma(Float64(Float64(alpha + beta) + 3.0), Float64(alpha + beta), fma(Float64(alpha + beta), 2.0, 6.0)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.52e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision] * N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 2.0 + 6.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.52e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in beta around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 94.9

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

   7. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 N/A

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

      23. metadata-eval 94.9

          \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, \color{blue}{6}\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   9. Applied rewrites 94.9%

      \[\leadsto \frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\color{blue}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right)} \cdot \left(\left(\beta + \alpha\right) + 2\right)} \]

   if 1.52e16 < beta

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\mathsf{fma}\left(\left(\alpha + \beta\right) + 3, \alpha + \beta, \mathsf{fma}\left(\alpha + \beta, 2, 6\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(t\_1 \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ 3.0 (+ beta alpha))))
   (if (<= beta 1.52e+16)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* (* t_1 t_0) t_0))
     (/ (/ (+ 1.0 alpha) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.52e+16) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = 3.0d0 + (beta + alpha)
    if (beta <= 1.52d+16) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / ((t_1 * t_0) * t_0)
    else
        tmp = ((1.0d0 + alpha) / t_1) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.52e+16) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((1.0 + alpha) / t_1) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = 3.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.52e+16:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0)
	else:
		tmp = ((1.0 + alpha) / t_1) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.52e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(Float64(t_1 * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = 3.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.52e+16)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / ((t_1 * t_0) * t_0);
	else
		tmp = ((1.0 + alpha) / t_1) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.52e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.52e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in beta around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.52e16 < beta

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.52 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 4.6e+16)
     (/ (/ (+ 1.0 beta) (fma (+ 5.0 beta) beta 6.0)) t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4.6e+16) {
		tmp = ((1.0 + beta) / fma((5.0 + beta), beta, 6.0)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 4.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / fma(Float64(5.0 + beta), beta, 6.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(5.0 + beta), $MachinePrecision] * beta + 6.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 68.1

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

   7. Applied rewrites 68.1%

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

   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 68.1%

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

   if 4.6e16 < beta

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.8

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

   7. Applied rewrites 82.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\mathsf{fma}\left(5 + \beta, \beta, 6\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.5)
     (/
      (fma
       (fma
        (fma 0.03780864197530864 beta -0.05092592592592592)
        beta
        0.027777777777777776)
       beta
       0.16666666666666666)
      t_0)
     (/ (/ (+ 1.0 alpha) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1.5) {
		tmp = fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / t_0;
	} else {
		tmp = ((1.0 + alpha) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.5], N[(N[(N[(N[(0.03780864197530864 * beta + -0.05092592592592592), $MachinePrecision] * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 67.2%

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

   if 1.5 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.0%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 72.2%

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

Alternative 6: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (/
    (fma
     (fma
      (fma 0.03780864197530864 beta -0.05092592592592592)
      beta
      0.027777777777777776)
     beta
     0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ (+ alpha beta) 2.0) 1.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (((alpha + beta) + 2.0) + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = Float64(fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / 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_] := If[LessEqual[beta, 2.2], N[(N[(N[(N[(0.03780864197530864 * beta + -0.05092592592592592), $MachinePrecision] * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $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 < 2.2000000000000002

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 67.2%

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

   if 2.2000000000000002 < beta

   1. Initial program 81.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03780864197530864, \beta, -0.05092592592592592\right), \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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 2.3)
     (/
      (fma
       (fma
        (fma 0.03780864197530864 beta -0.05092592592592592)
        beta
        0.027777777777777776)
       beta
       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 <= 2.3) {
		tmp = fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 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 <= 2.3)
		tmp = Float64(fma(fma(fma(0.03780864197530864, beta, -0.05092592592592592), beta, 0.027777777777777776), beta, 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, 2.3], N[(N[(N[(N[(0.03780864197530864 * beta + -0.05092592592592592), $MachinePrecision] * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 67.2%

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

   if 2.2999999999999998 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.0%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 8: 97.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t\_0}\\ \end{array} \end{array} \]

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 1.9)
     (/
      (fma
       (fma -0.05092592592592592 beta 0.027777777777777776)
       beta
       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 <= 1.9) {
		tmp = fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 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 <= 1.9)
		tmp = Float64(fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 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, 1.9], N[(N[(N[(-0.05092592592592592 * beta + 0.027777777777777776), $MachinePrecision] * beta + 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 < 1.8999999999999999

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 67.1%

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

   if 1.8999999999999999 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.0%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 9: 97.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.05092592592592592, \beta, 0.027777777777777776\right), \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (/
    (fma
     (fma -0.05092592592592592 beta 0.027777777777777776)
     beta
     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 <= 2.1) {
		tmp = fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 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 <= 2.1)
		tmp = Float64(fma(fma(-0.05092592592592592, beta, 0.027777777777777776), beta, 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, 2.1], N[(N[(N[(-0.05092592592592592 * beta + 0.027777777777777776), $MachinePrecision] * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 67.1%

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

   if 2.10000000000000009 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 81.6%

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

Alternative 10: 96.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, \beta, 0.16666666666666666\right)}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/
    (fma 0.027777777777777776 beta 0.16666666666666666)
    (+ (+ beta alpha) 2.0))
   (if (<= beta 1.5e+151)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = fma(0.027777777777777776, beta, 0.16666666666666666) / ((beta + alpha) + 2.0);
	} else if (beta <= 1.5e+151) {
		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 <= 5.2)
		tmp = Float64(fma(0.027777777777777776, beta, 0.16666666666666666) / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 1.5e+151)
		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, 5.2], N[(N[(0.027777777777777776 * beta + 0.16666666666666666), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+151], 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 < 5.20000000000000018

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 66.8%

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

   if 5.20000000000000018 < beta < 1.5e151

   1. Initial program 85.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 1.5e151 < beta

   1. Initial program 77.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 89.1%

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

Alternative 11: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{0.16666666666666666}{\left(\beta + \alpha\right) + 2}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 0.16666666666666666 (+ (+ beta alpha) 2.0))
   (if (<= beta 1.5e+151)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 1.5e+151) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else if (beta <= 1.5d+151) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.0) {
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	} else if (beta <= 1.5e+151) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 8.0:
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0)
	elif beta <= 1.5e+151:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.0)
		tmp = Float64(0.16666666666666666 / Float64(Float64(beta + alpha) + 2.0));
	elseif (beta <= 1.5e+151)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	elseif (beta <= 1.5e+151)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+151], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 66.3%

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

   if 8 < beta < 1.5e151

   1. Initial program 85.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 1.5e151 < beta

   1. Initial program 77.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 89.1%

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

Alternative 12: 97.1% accurate, 2.6× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/
    (fma 0.027777777777777776 beta 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 <= 5.2) {
		tmp = fma(0.027777777777777776, beta, 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 <= 5.2)
		tmp = Float64(fma(0.027777777777777776, beta, 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, 5.2], N[(N[(0.027777777777777776 * beta + 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 < 5.20000000000000018

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 66.8%

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

   if 5.20000000000000018 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 81.6%

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

Alternative 13: 93.6% accurate, 3.2× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.0)
   (/ 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 <= 8.0) {
		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 <= 8.0d0) 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 <= 8.0) {
		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 <= 8.0:
		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 <= 8.0)
		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 <= 8.0)
		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, 8.0], 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

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 66.3%

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

   if 8 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

Alternative 14: 90.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.0d0) then
        tmp = 0.16666666666666666d0 / ((beta + alpha) + 2.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.0)
		tmp = 0.16666666666666666 / ((beta + alpha) + 2.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 8.0], N[(0.16666666666666666 / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 67.7

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 66.3%

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

   if 8 < beta

   1. Initial program 81.0%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 78.1%

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

Alternative 15: 52.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.49:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.49) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.49) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.49d0) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.49) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 0.49:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.49)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.49)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 0.49], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.48999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 35.6%

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

   if 0.48999999999999999 < alpha

   1. Initial program 80.8%

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

   3. 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 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 15.5%

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

Alternative 16: 31.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 29.3

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

5. Applied rewrites 29.3%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 20.2%

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

Reproduce

herbie shell --seed 2024313 
(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)))