Octave 3.8, jcobi/3

Percentage Accurate: 93.9% → 99.5%

Time: 12.5s

Alternatives: 18

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: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0} \cdot \frac{1}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1e+126)
     (*
      (/ (+ (fma alpha beta (+ beta alpha)) 1.0) t_0)
      (/ 1.0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 1e+126) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / t_0) * (1.0 / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 1e+126)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / t_0) * Float64(1.0 / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	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_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1e+126], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999925e125

   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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.0%

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

   if 9.99999999999999925e125 < beta

   1. Initial program 80.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 93.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2} \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0 \cdot \left(\left(\beta + \alpha\right) + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2e+114)
     (/
      (/
       (+ (fma alpha beta (+ beta alpha)) 1.0)
       (* t_0 (+ (+ beta alpha) 3.0)))
      t_0)
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2e+114) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / (t_0 * ((beta + alpha) + 3.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2e+114)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_0 * Float64(Float64(beta + alpha) + 3.0))) / t_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_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+114], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e114

   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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 98.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if 2e114 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 93.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0 \cdot t\_0}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 2e+114)
     (/
      (/ (+ (fma alpha beta (+ beta alpha)) 1.0) (* t_0 t_0))
      (+ alpha (+ beta 3.0)))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 2e+114) {
		tmp = ((fma(alpha, beta, (beta + alpha)) + 1.0) / (t_0 * t_0)) / (alpha + (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 2e+114)
		tmp = Float64(Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_0 * t_0)) / Float64(alpha + Float64(beta + 3.0)));
	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_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2e+114], N[(N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e114

   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

   4. Applied rewrites 98.0%

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

   if 2e114 < beta

   1. Initial program 81.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 93.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{t\_1 \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_1}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ alpha (+ beta 3.0))))
   (if (<= beta 9e+99)
     (/ (* (+ beta 1.0) (+ alpha 1.0)) (* t_1 (* t_0 t_0)))
     (/ (/ (+ alpha 1.0) t_1) (+ alpha (+ beta 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 9e+99) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_1) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = alpha + (beta + 3.0d0)
    if (beta <= 9d+99) then
        tmp = ((beta + 1.0d0) * (alpha + 1.0d0)) / (t_1 * (t_0 * t_0))
    else
        tmp = ((alpha + 1.0d0) / t_1) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 9e+99) {
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / t_1) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 9e+99:
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * (t_0 * t_0))
	else:
		tmp = ((alpha + 1.0) / t_1) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 9e+99)
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(alpha + 1.0)) / Float64(t_1 * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_1) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 9e+99)
		tmp = ((beta + 1.0) * (alpha + 1.0)) / (t_1 * (t_0 * t_0));
	else
		tmp = ((alpha + 1.0) / t_1) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e+99], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.9999999999999999e99

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in alpha around 0

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

      1. distribute-lft-out N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-+r+ N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 95.2

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

   7. Applied rewrites 95.2%

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

   if 8.9999999999999999e99 < beta

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.5

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

   7. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 88.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \beta \cdot \left(\alpha + 1\right)}{t\_0 \cdot \left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 2e+37)
     (/
      (+ 1.0 (* beta (+ alpha 1.0)))
      (* t_0 (* (+ (+ beta alpha) 2.0) (+ beta 2.0))))
     (/ (/ (+ alpha 1.0) t_0) (+ alpha (+ beta 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2e+37) {
		tmp = (1.0 + (beta * (alpha + 1.0))) / (t_0 * (((beta + alpha) + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    if (beta <= 2d+37) then
        tmp = (1.0d0 + (beta * (alpha + 1.0d0))) / (t_0 * (((beta + alpha) + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / t_0) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 2e+37) {
		tmp = (1.0 + (beta * (alpha + 1.0))) / (t_0 * (((beta + alpha) + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 2e+37:
		tmp = (1.0 + (beta * (alpha + 1.0))) / (t_0 * (((beta + alpha) + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / t_0) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 2e+37)
		tmp = Float64(Float64(1.0 + Float64(beta * Float64(alpha + 1.0))) / Float64(t_0 * Float64(Float64(Float64(beta + alpha) + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 2e+37)
		tmp = (1.0 + (beta * (alpha + 1.0))) / (t_0 * (((beta + alpha) + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / t_0) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+37], N[(N[(1.0 + N[(beta * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999991e37

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.7

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 86.0

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

   10. Applied rewrites 86.0%

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

   if 1.99999999999999991e37 < beta

   1. Initial program 83.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing
   3. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 88.9

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

   7. Applied rewrites 88.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 87.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \beta \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{t\_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 4.9e+33)
     (/
      (+ (fma alpha beta (+ beta alpha)) 1.0)
      (* t_0 (* (+ beta 2.0) (+ beta 2.0))))
     (/ (/ (+ alpha 1.0) t_0) (+ alpha (+ beta 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 4.9e+33) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / (t_0 * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 4.9e+33)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(t_0 * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(alpha + Float64(beta + 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[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.9e+33], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.90000000000000014e33

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.2

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

   7. Applied rewrites 67.2%

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

   if 4.90000000000000014e33 < beta

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 85.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.9e+33)
   (/
    (+ (fma alpha beta (+ beta alpha)) 1.0)
    (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.9e+33) {
		tmp = (fma(alpha, beta, (beta + alpha)) + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.9e+33)
		tmp = Float64(Float64(fma(alpha, beta, Float64(beta + alpha)) + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 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, 4.9e+33], N[(N[(N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.90000000000000014e33

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if 4.90000000000000014e33 < beta

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 85.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right) + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+32)
   (/ (+ beta 1.0) (fma beta (fma beta (+ beta 7.0) 16.0) 12.0))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 3.0))) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+32) {
		tmp = (beta + 1.0) / fma(beta, fma(beta, (beta + 7.0), 16.0), 12.0);
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 3.0))) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+32)
		tmp = Float64(Float64(beta + 1.0) / fma(beta, fma(beta, Float64(beta + 7.0), 16.0), 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 3.0))) / Float64(alpha + Float64(beta + 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, 6e+32], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta * N[(beta + 7.0), $MachinePrecision] + 16.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e32

   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. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 67.1%

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

   if 6e32 < beta

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 89.0

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

   7. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 85.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.8e+33)
   (/ (+ beta 1.0) (fma beta (fma beta (+ beta 7.0) 16.0) 12.0))
   (/ (/ (+ alpha 1.0) beta) (+ (+ beta alpha) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.8e+33) {
		tmp = (beta + 1.0) / fma(beta, fma(beta, (beta + 7.0), 16.0), 12.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / ((beta + alpha) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.8e+33)
		tmp = Float64(Float64(beta + 1.0) / fma(beta, fma(beta, Float64(beta + 7.0), 16.0), 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(Float64(beta + alpha) + 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, 7.8e+33], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta * N[(beta + 7.0), $MachinePrecision] + 16.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.8000000000000004e33

   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. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 67.1%

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

   if 7.8000000000000004e33 < beta

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. 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. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 79.5

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.6

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

   9. Applied rewrites 85.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+35}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.55e+35)
   (/ (+ beta 1.0) (fma beta (fma beta (+ beta 7.0) 16.0) 12.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.55e+35) {
		tmp = (beta + 1.0) / fma(beta, fma(beta, (beta + 7.0), 16.0), 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 <= 2.55e+35)
		tmp = Float64(Float64(beta + 1.0) / fma(beta, fma(beta, Float64(beta + 7.0), 16.0), 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, 2.55e+35], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta * N[(beta + 7.0), $MachinePrecision] + 16.0), $MachinePrecision] + 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 < 2.55000000000000009e35

   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. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 67.1%

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

   if 2.55000000000000009e35 < beta

   1. Initial program 83.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 85.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+35}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 68.2%

      \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\alpha + \left(\beta + 3\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      11. lift-+.f64 68.2

          \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   10. Applied rewrites 68.2%

       \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   if 6.20000000000000018 < beta

   1. Initial program 85.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ alpha (+ beta 3.0)))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 68.2%

      \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\alpha + \left(\beta + 3\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      11. lift-+.f64 68.2

          \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   10. Applied rewrites 68.2%

       \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   if 6.20000000000000018 < beta

   1. Initial program 85.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2) (/ 0.25 (+ alpha (+ beta 3.0))) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (alpha + (beta + 3.0d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (alpha + (beta + 3.0));
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (alpha + (beta + 3.0))
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (alpha + (beta + 3.0));
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 68.2%

      \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(\alpha + \beta\right) + \color{blue}{3}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\alpha + \left(\beta + 3\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

      11. lift-+.f64 68.2

          \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right)} + \alpha} \]

   10. Applied rewrites 68.2%

       \[\leadsto \frac{0.25}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

   if 6.20000000000000018 < beta

   1. Initial program 85.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65)
   (fma
    beta
    (fma beta -0.011574074074074073 -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = fma(beta, fma(beta, -0.011574074074074073, -0.027777777777777776), 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 <= 1.65)
		tmp = fma(beta, fma(beta, -0.011574074074074073, -0.027777777777777776), 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, 1.65], N[(beta * N[(beta * -0.011574074074074073 + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999

   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. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 66.5

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

   5. Applied rewrites 66.5%

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

   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 66.5%

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

   if 1.6499999999999999 < beta

   1. Initial program 85.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 78.4%

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

Alternative 15: 73.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.05e+71) (/ 0.25 (+ beta 3.0)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05e+71) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.05d+71) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05e+71) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.05e+71:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05e+71)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05e+71)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.05e+71], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0500000000000001e71

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 62.4%

      \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{3 + \beta}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\beta + 3}} \]

       2. lower-+.f64 60.7

          \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   11. Applied rewrites 60.7%

       \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   if 2.0500000000000001e71 < beta

   1. Initial program 81.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 66.7%

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

Alternative 16: 45.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

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 = 0.25d0 / (beta + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 73.2

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

5. Applied rewrites 73.2%

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

6. Taylor expanded in beta around 0

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

8. Applied rewrites 47.0%

   \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

9. Taylor expanded in alpha around 0

   \[\leadsto \frac{\frac{1}{4}}{\color{blue}{3 + \beta}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\beta + 3}} \]

    2. lower-+.f64 45.7

       \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

11. Applied rewrites 45.7%

    \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]
12. Add Preprocessing

Alternative 17: 44.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\alpha + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ alpha 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

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 = 0.25d0 / (alpha + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (alpha + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(alpha + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (alpha + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 73.2

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

5. Applied rewrites 73.2%

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

6. Taylor expanded in beta around 0

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

8. Applied rewrites 47.0%

   \[\leadsto \frac{0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

9. Taylor expanded in beta around 0

   \[\leadsto \frac{\frac{1}{4}}{\color{blue}{3 + \alpha}} \]
10. Step-by-step derivation

    1. lower-+.f64 46.0

       \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

11. Applied rewrites 46.0%

    \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

12. Final simplification 46.0%

    \[\leadsto \frac{0.25}{\alpha + 3} \]
13. Add Preprocessing

Alternative 18: 43.8% 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 94.8%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
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. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 69.0

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

5. Applied rewrites 69.0%

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

6. Taylor expanded in beta around 0

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

8. Applied rewrites 44.6%

   \[\leadsto 0.08333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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)))