Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.7%

Time: 11.3s

Alternatives: 19

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{t\_0}}{\left(\left(3 + \beta\right) + \alpha\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 alpha) beta)))
   (if (<= beta 3.7e+105)
     (/
      (/ (fma (+ alpha 1.0) beta (+ alpha 1.0)) t_0)
      (* (+ (+ 3.0 beta) alpha) t_0))
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (+ 2.0 alpha) beta) (+ alpha 1.0)))
       (+ (+ alpha beta) 2.0))
      (+ (+ (+ alpha beta) 1.0) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 3.7e+105) {
		tmp = (fma((alpha + 1.0), beta, (alpha + 1.0)) / t_0) / (((3.0 + beta) + alpha) * t_0);
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - (((2.0 + alpha) / beta) * (alpha + 1.0))) / ((alpha + beta) + 2.0)) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 3.7e+105)
		tmp = Float64(Float64(fma(Float64(alpha + 1.0), beta, Float64(alpha + 1.0)) / t_0) / Float64(Float64(Float64(3.0 + beta) + alpha) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(Float64(2.0 + alpha) / beta) * Float64(alpha + 1.0))) / Float64(Float64(alpha + beta) + 2.0)) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 3.7e+105], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999985e105

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.7%

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

   if 3.69999999999999985e105 < beta

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.6

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 77.7

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

   7. Applied rewrites 77.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{\left(2 + \alpha\right) + \beta}}{\left(\left(3 + \beta\right) + \alpha\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{2 + \alpha}{\beta} \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{t\_0}}{\left(\left(3 + \beta\right) + \alpha\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(\alpha + 1\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 alpha) beta)))
   (if (<= beta 3.7e+105)
     (/
      (/ (fma (+ alpha 1.0) beta (+ alpha 1.0)) t_0)
      (* (+ (+ 3.0 beta) alpha) t_0))
     (/
      (/
       (-
        (+ (/ alpha beta) (+ (+ (/ 1.0 beta) alpha) 1.0))
        (* (/ (fma 2.0 alpha 4.0) beta) (+ alpha 1.0)))
       beta)
      (+ (+ (+ alpha beta) 1.0) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 3.7e+105) {
		tmp = (fma((alpha + 1.0), beta, (alpha + 1.0)) / t_0) / (((3.0 + beta) + alpha) * t_0);
	} else {
		tmp = ((((alpha / beta) + (((1.0 / beta) + alpha) + 1.0)) - ((fma(2.0, alpha, 4.0) / beta) * (alpha + 1.0))) / beta) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 3.7e+105)
		tmp = Float64(Float64(fma(Float64(alpha + 1.0), beta, Float64(alpha + 1.0)) / t_0) / Float64(Float64(Float64(3.0 + beta) + alpha) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha / beta) + Float64(Float64(Float64(1.0 / beta) + alpha) + 1.0)) - Float64(Float64(fma(2.0, alpha, 4.0) / beta) * Float64(alpha + 1.0))) / beta) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 3.7e+105], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha / beta), $MachinePrecision] + N[(N[(N[(1.0 / beta), $MachinePrecision] + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999985e105

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.7%

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

   if 3.69999999999999985e105 < beta

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 67.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.6

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 77.7

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

   7. Applied rewrites 77.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{\left(2 + \alpha\right) + \beta}}{\left(\left(3 + \beta\right) + \alpha\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\beta} + \left(\left(\frac{1}{\beta} + \alpha\right) + 1\right)\right) - \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} \cdot \left(\alpha + 1\right)}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(3 + \beta\right) + \alpha\\ t_1 := \left(2 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{t\_1}}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0} \cdot \left(\alpha + 1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 3.0 beta) alpha)) (t_1 (+ (+ 2.0 alpha) beta)))
   (if (<= beta 3.7e+105)
     (/ (/ (fma (+ alpha 1.0) beta (+ alpha 1.0)) t_1) (* t_0 t_1))
     (/ (* (/ 1.0 t_0) (+ alpha 1.0)) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (3.0 + beta) + alpha;
	double t_1 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 3.7e+105) {
		tmp = (fma((alpha + 1.0), beta, (alpha + 1.0)) / t_1) / (t_0 * t_1);
	} else {
		tmp = ((1.0 / t_0) * (alpha + 1.0)) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(3.0 + beta) + alpha)
	t_1 = Float64(Float64(2.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 3.7e+105)
		tmp = Float64(Float64(fma(Float64(alpha + 1.0), beta, Float64(alpha + 1.0)) / t_1) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(1.0 / t_0) * Float64(alpha + 1.0)) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 3.7e+105], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] * beta + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.69999999999999985e105

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 98.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.7%

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

   if 3.69999999999999985e105 < beta

   1. Initial program 67.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

   7. Applied rewrites 78.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha + 1, \beta, \alpha + 1\right)}{\left(2 + \alpha\right) + \beta}}{\left(\left(3 + \beta\right) + \alpha\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(3 + \beta\right) + \alpha} \cdot \left(\alpha + 1\right)}{\left(2 + \alpha\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) + 1}{\left(t\_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(3 + \beta\right) + \alpha} \cdot \left(\alpha + 1\right)}{\left(2 + \alpha\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 8e+97)
     (/
      (+ (fma beta alpha (+ alpha beta)) 1.0)
      (* (* t_0 (+ (+ alpha beta) 3.0)) t_0))
     (/
      (* (/ 1.0 (+ (+ 3.0 beta) alpha)) (+ alpha 1.0))
      (+ (+ 2.0 alpha) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 8e+97) {
		tmp = (fma(beta, alpha, (alpha + beta)) + 1.0) / ((t_0 * ((alpha + beta) + 3.0)) * t_0);
	} else {
		tmp = ((1.0 / ((3.0 + beta) + alpha)) * (alpha + 1.0)) / ((2.0 + alpha) + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 8e+97)
		tmp = Float64(Float64(fma(beta, alpha, Float64(alpha + beta)) + 1.0) / Float64(Float64(t_0 * Float64(Float64(alpha + beta) + 3.0)) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(3.0 + beta) + alpha)) * Float64(alpha + 1.0)) / Float64(Float64(2.0 + alpha) + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 8e+97], N[(N[(N[(beta * alpha + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.0000000000000006e97

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   if 8.0000000000000006e97 < beta

   1. Initial program 68.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

   7. Applied rewrites 79.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(3 + \beta\right) + \alpha} \cdot \left(\alpha + 1\right)}{\left(2 + \alpha\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(\alpha + 1\right)}{\left(t\_0 \cdot t\_0\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(3 + \beta\right) + \alpha} \cdot \left(\alpha + 1\right)}{\left(2 + \alpha\right) + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 8e+97)
     (/ (* (+ 1.0 beta) (+ alpha 1.0)) (* (* t_0 t_0) (+ (+ alpha beta) 3.0)))
     (/
      (* (/ 1.0 (+ (+ 3.0 beta) alpha)) (+ alpha 1.0))
      (+ (+ 2.0 alpha) beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 8e+97) {
		tmp = ((1.0 + beta) * (alpha + 1.0)) / ((t_0 * t_0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((1.0 / ((3.0 + beta) + alpha)) * (alpha + 1.0)) / ((2.0 + alpha) + beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    if (beta <= 8d+97) then
        tmp = ((1.0d0 + beta) * (alpha + 1.0d0)) / ((t_0 * t_0) * ((alpha + beta) + 3.0d0))
    else
        tmp = ((1.0d0 / ((3.0d0 + beta) + alpha)) * (alpha + 1.0d0)) / ((2.0d0 + alpha) + beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 8e+97) {
		tmp = ((1.0 + beta) * (alpha + 1.0)) / ((t_0 * t_0) * ((alpha + beta) + 3.0));
	} else {
		tmp = ((1.0 / ((3.0 + beta) + alpha)) * (alpha + 1.0)) / ((2.0 + alpha) + beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	tmp = 0
	if beta <= 8e+97:
		tmp = ((1.0 + beta) * (alpha + 1.0)) / ((t_0 * t_0) * ((alpha + beta) + 3.0))
	else:
		tmp = ((1.0 / ((3.0 + beta) + alpha)) * (alpha + 1.0)) / ((2.0 + alpha) + beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 8e+97)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(alpha + 1.0)) / Float64(Float64(t_0 * t_0) * Float64(Float64(alpha + beta) + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(3.0 + beta) + alpha)) * Float64(alpha + 1.0)) / Float64(Float64(2.0 + alpha) + beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	tmp = 0.0;
	if (beta <= 8e+97)
		tmp = ((1.0 + beta) * (alpha + 1.0)) / ((t_0 * t_0) * ((alpha + beta) + 3.0));
	else
		tmp = ((1.0 / ((3.0 + beta) + alpha)) * (alpha + 1.0)) / ((2.0 + alpha) + beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 8e+97], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.0000000000000006e97

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   8. Applied rewrites 93.9%

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

   if 8.0000000000000006e97 < beta

   1. Initial program 68.6%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

   7. Applied rewrites 79.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(\alpha + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(3 + \beta\right) + \alpha} \cdot \left(\alpha + 1\right)}{\left(2 + \alpha\right) + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) + \beta}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.4)
   (/ (/ (+ alpha 1.0) (* (+ 2.0 alpha) (+ 2.0 alpha))) (+ (+ alpha beta) 3.0))
   (/ (/ (+ alpha 1.0) (+ (+ 2.0 alpha) beta)) (+ (+ 3.0 beta) alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = ((alpha + 1.0) / ((2.0 + alpha) * (2.0 + alpha))) / ((alpha + beta) + 3.0);
	} else {
		tmp = ((alpha + 1.0) / ((2.0 + alpha) + beta)) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.4d0) then
        tmp = ((alpha + 1.0d0) / ((2.0d0 + alpha) * (2.0d0 + alpha))) / ((alpha + beta) + 3.0d0)
    else
        tmp = ((alpha + 1.0d0) / ((2.0d0 + alpha) + beta)) / ((3.0d0 + beta) + alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.4) {
		tmp = ((alpha + 1.0) / ((2.0 + alpha) * (2.0 + alpha))) / ((alpha + beta) + 3.0);
	} else {
		tmp = ((alpha + 1.0) / ((2.0 + alpha) + beta)) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.4:
		tmp = ((alpha + 1.0) / ((2.0 + alpha) * (2.0 + alpha))) / ((alpha + beta) + 3.0)
	else:
		tmp = ((alpha + 1.0) / ((2.0 + alpha) + beta)) / ((3.0 + beta) + alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.4)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(2.0 + alpha) * Float64(2.0 + alpha))) / Float64(Float64(alpha + beta) + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(2.0 + alpha) + beta)) / Float64(Float64(3.0 + beta) + alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.4)
		tmp = ((alpha + 1.0) / ((2.0 + alpha) * (2.0 + alpha))) / ((alpha + beta) + 3.0);
	else
		tmp = ((alpha + 1.0) / ((2.0 + alpha) + beta)) / ((3.0 + beta) + alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.4], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999991

   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

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 98.8

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

   7. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 98.8

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

   9. Applied rewrites 98.8%

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

   if 2.39999999999999991 < beta

   1. Initial program 74.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 79.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) + \beta}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) + \beta}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5e+14)
   (/ (+ 1.0 beta) (* (fma (+ 4.0 beta) beta 4.0) (+ 3.0 beta)))
   (/ (/ (+ alpha 1.0) (+ (+ 2.0 alpha) beta)) (+ (+ 3.0 beta) alpha))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5e+14) {
		tmp = (1.0 + beta) / (fma((4.0 + beta), beta, 4.0) * (3.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / ((2.0 + alpha) + beta)) / ((3.0 + beta) + alpha);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5e+14)
		tmp = Float64(Float64(1.0 + beta) / Float64(fma(Float64(4.0 + beta), beta, 4.0) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(2.0 + alpha) + beta)) / Float64(Float64(3.0 + beta) + alpha));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(4.0 + beta), $MachinePrecision] * beta + 4.0), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e14

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 69.7%

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

   if 3.5e14 < beta

   1. Initial program 72.4%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 80.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(2 + \alpha\right) + \beta}}{\left(3 + \beta\right) + \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+14)
   (/ (+ 1.0 beta) (* (fma (+ 4.0 beta) beta 4.0) (+ 3.0 beta)))
   (/ (/ (+ alpha 1.0) beta) (+ (+ (+ alpha beta) 1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+14) {
		tmp = (1.0 + beta) / (fma((4.0 + beta), beta, 4.0) * (3.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / (((alpha + beta) + 1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+14)
		tmp = Float64(Float64(1.0 + beta) / Float64(fma(Float64(4.0 + beta), beta, 4.0) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(Float64(alpha + beta) + 1.0) + 2.0));
	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, 3.8e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(4.0 + beta), $MachinePrecision] * beta + 4.0), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e14

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 69.7%

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

   if 3.8e14 < beta

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 72.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.4

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 80.3

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

   7. Applied rewrites 80.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\left(\alpha + \beta\right) + 1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+14)
   (/ (+ 1.0 beta) (* (fma (+ 4.0 beta) beta 4.0) (+ 3.0 beta)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+14) {
		tmp = (1.0 + beta) / (fma((4.0 + beta), beta, 4.0) * (3.0 + beta));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+14)
		tmp = Float64(Float64(1.0 + beta) / Float64(fma(Float64(4.0 + beta), beta, 4.0) * Float64(3.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(4.0 + beta), $MachinePrecision] * beta + 4.0), $MachinePrecision] * N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e14

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 69.7%

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

   if 3.8e14 < beta

   1. Initial program 72.4%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 80.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(4 + \beta, \beta, 4\right) \cdot \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \beta, \beta, 16\right), \beta, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8e+14)
   (/ (+ 1.0 beta) (fma (fma (+ 7.0 beta) beta 16.0) beta 12.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8e+14) {
		tmp = (1.0 + beta) / fma(fma((7.0 + beta), beta, 16.0), beta, 12.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8e+14)
		tmp = Float64(Float64(1.0 + beta) / fma(fma(Float64(7.0 + beta), beta, 16.0), beta, 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.8e+14], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(N[(7.0 + beta), $MachinePrecision] * beta + 16.0), $MachinePrecision] * beta + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e14

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\beta + 1}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.7%

      \[\leadsto \frac{\beta + 1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \beta, \beta, 16\right), \color{blue}{\beta}, 12\right)} \]

   if 3.8e14 < beta

   1. Initial program 72.4%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 80.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \beta}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \beta, \beta, 16\right), \beta, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\alpha + 1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ (+ alpha 1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = (alpha + 1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(Float64(alpha + 1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\alpha + 1}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.6%

      \[\leadsto \frac{\alpha + 1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 4.79999999999999982 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 78.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\alpha + 1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (fma
    (fma
     (fma 0.024691358024691357 beta -0.011574074074074073)
     beta
     -0.027777777777777776)
    beta
    0.08333333333333333)
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(N[(N[(0.024691358024691357 * beta + -0.011574074074074073), $MachinePrecision] * beta + -0.027777777777777776), $MachinePrecision] * beta + 0.08333333333333333), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \color{blue}{\beta}, 0.08333333333333333\right) \]

   if 2.2000000000000002 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 78.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (fma
    (fma
     (fma 0.024691358024691357 beta -0.011574074074074073)
     beta
     -0.027777777777777776)
    beta
    0.08333333333333333)
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(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, 2.2], N[(N[(N[(0.024691358024691357 * beta + -0.011574074074074073), $MachinePrecision] * beta + -0.027777777777777776), $MachinePrecision] * beta + 0.08333333333333333), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \color{blue}{\beta}, 0.08333333333333333\right) \]

   if 2.2000000000000002 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

Alternative 14: 91.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.1)
   (fma
    (fma
     (fma 0.024691358024691357 beta -0.011574074074074073)
     beta
     -0.027777777777777776)
    beta
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = fma(fma(fma(0.024691358024691357, beta, -0.011574074074074073), beta, -0.027777777777777776), beta, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(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[(0.024691358024691357 * beta + -0.011574074074074073), $MachinePrecision] * beta + -0.027777777777777776), $MachinePrecision] * beta + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $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. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \beta, -0.011574074074074073\right), \beta, -0.027777777777777776\right), \color{blue}{\beta}, 0.08333333333333333\right) \]

   if 2.10000000000000009 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 72.7%

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

Alternative 15: 91.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7)
   (fma
    (fma -0.011574074074074073 beta -0.027777777777777776)
    beta
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = fma(fma(-0.011574074074074073, beta, -0.027777777777777776), beta, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = fma(fma(-0.011574074074074073, beta, -0.027777777777777776), beta, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(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, 1.7], N[(N[(-0.011574074074074073 * beta + -0.027777777777777776), $MachinePrecision] * beta + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \beta, -0.027777777777777776\right), \color{blue}{\beta}, 0.08333333333333333\right) \]

   if 1.69999999999999996 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 72.7%

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

Alternative 16: 74.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \beta, -0.027777777777777776\right), \beta, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.75)
   (fma
    (fma -0.011574074074074073 beta -0.027777777777777776)
    beta
    0.08333333333333333)
   (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.75) {
		tmp = fma(fma(-0.011574074074074073, beta, -0.027777777777777776), beta, 0.08333333333333333);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.75)
		tmp = fma(fma(-0.011574074074074073, beta, -0.027777777777777776), beta, 0.08333333333333333);
	else
		tmp = Float64(alpha / Float64(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, 1.75], N[(N[(-0.011574074074074073 * beta + -0.027777777777777776), $MachinePrecision] * beta + 0.08333333333333333), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75

   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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \beta, -0.027777777777777776\right), \color{blue}{\beta}, 0.08333333333333333\right) \]

   if 1.75 < beta

   1. Initial program 74.0%

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

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 46.4%

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

Alternative 17: 45.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma
  (fma -0.011574074074074073 alpha -0.027777777777777776)
  alpha
  0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 46.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]
9. Add Preprocessing

Alternative 18: 45.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(-0.027777777777777776, \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma -0.027777777777777776 alpha 0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(-0.027777777777777776, alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(-0.027777777777777776, alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(-0.027777777777777776 * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \alpha} \]
7. Step-by-step derivation

8. Applied rewrites 46.6%

   \[\leadsto \mathsf{fma}\left(-0.027777777777777776, \color{blue}{\alpha}, 0.08333333333333333\right) \]
9. Add Preprocessing

Alternative 19: 45.5% accurate, 84.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

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 = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in beta around 0

   \[\leadsto \frac{1}{12} \]
7. Step-by-step derivation

8. Applied rewrites 47.0%

   \[\leadsto 0.08333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))