Octave 3.8, jcobi/3

Percentage Accurate: 94.0% → 99.6%

Time: 13.1s

Alternatives: 14

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\left(t\_0 \cdot \frac{t\_0}{1 + \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 3e+140)
     (/
      1.0
      (*
       (* t_0 (/ t_0 (+ 1.0 (fma alpha beta (+ beta alpha)))))
       (+ alpha (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 3e+140) {
		tmp = 1.0 / ((t_0 * (t_0 / (1.0 + fma(alpha, beta, (beta + alpha))))) * (alpha + (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 3e+140)
		tmp = Float64(1.0 / Float64(Float64(t_0 * Float64(t_0 / Float64(1.0 + fma(alpha, beta, Float64(beta + alpha))))) * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 3e+140], N[(1.0 / N[(N[(t$95$0 * N[(t$95$0 / N[(1.0 + N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.99999999999999997e140

   1. Initial program 98.5%

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

   4. Applied rewrites 98.1%

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

   if 2.99999999999999997e140 < beta

   1. Initial program 76.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \frac{\left(\beta + \alpha\right) + 2}{1 + \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 9.5e+17)
     (/
      (+ 1.0 (fma alpha beta (+ beta alpha)))
      (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ 1.0 alpha) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 9.5e+17) {
		tmp = (1.0 + fma(alpha, beta, (beta + alpha))) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((1.0 + alpha) / 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 <= 9.5e+17)
		tmp = Float64(Float64(1.0 + fma(alpha, beta, Float64(beta + alpha))) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9.5e+17], N[(N[(1.0 + N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e17

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

   4. Applied rewrites 94.2%

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

   if 9.5e17 < beta

   1. Initial program 83.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.3e+17)
   (/
    (/ (+ beta 1.0) (fma beta (+ beta 4.0) 4.0))
    (+ 1.0 (+ (+ beta alpha) 2.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.3e+17) {
		tmp = ((beta + 1.0) / fma(beta, (beta + 4.0), 4.0)) / (1.0 + ((beta + alpha) + 2.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.3e+17)
		tmp = Float64(Float64(Float64(beta + 1.0) / fma(beta, Float64(beta + 4.0), 4.0)) / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.3e+17], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.3e17

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 71.3

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

   8. Applied rewrites 71.3%

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

   if 4.3e17 < beta

   1. Initial program 83.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+19}:\\ \;\;\;\;\left(-1 - \beta\right) \cdot \frac{1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(-3 - \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+19)
   (* (- -1.0 beta) (/ 1.0 (* (fma beta (+ beta 4.0) 4.0) (- -3.0 beta))))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+19) {
		tmp = (-1.0 - beta) * (1.0 / (fma(beta, (beta + 4.0), 4.0) * (-3.0 - beta)));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+19)
		tmp = Float64(Float64(-1.0 - beta) * Float64(1.0 / Float64(fma(beta, Float64(beta + 4.0), 4.0) * Float64(-3.0 - beta))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6e+19], N[(N[(-1.0 - beta), $MachinePrecision] * N[(1.0 / N[(N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6e19

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 71.4

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

   8. Applied rewrites 71.4%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-+.f64 70.3

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

   11. Applied rewrites 70.3%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. frac-2neg N/A

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

       5. lift-+.f64 N/A

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

       6. associate-/l/ N/A

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

       7. div-inv N/A

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

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

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

       9. lower-*.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. unsub-neg N/A

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

       14. lower--.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. *-commutative N/A

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

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

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

       18. lower-*.f64 N/A

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

       19. lift-+.f64 N/A

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

   13. Applied rewrites 70.3%

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

   if 6e19 < 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 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 5: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \mathsf{fma}\left(\beta, \beta + 4, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6e+16)
   (/ (+ beta 1.0) (* (+ beta 3.0) (fma beta (+ beta 4.0) 4.0)))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6e+16) {
		tmp = (beta + 1.0) / ((beta + 3.0) * fma(beta, (beta + 4.0), 4.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * fma(beta, Float64(beta + 4.0), 4.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e16

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 70.2

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

   8. Applied rewrites 70.2%

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

   if 4.6e16 < beta

   1. Initial program 83.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \mathsf{fma}\left(\beta, \beta + 4, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6e+16)
   (/ (+ beta 1.0) (fma beta (fma beta (+ beta 7.0) 16.0) 12.0))
   (/ (/ (+ 1.0 alpha) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6e+16) {
		tmp = (beta + 1.0) / fma(beta, fma(beta, (beta + 7.0), 16.0), 12.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6e+16)
		tmp = Float64(Float64(beta + 1.0) / fma(beta, fma(beta, Float64(beta + 7.0), 16.0), 12.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.6e+16], 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[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e16

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 70.2

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

   8. Applied rewrites 70.2%

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

   if 4.6e16 < beta

   1. Initial program 83.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2)
   (fma
    beta
    (fma
     beta
     (fma beta 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (if (<= beta 1.35e+154)
     (/ (+ 1.0 alpha) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(beta, fma(beta, fma(beta, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else if (beta <= 1.35e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2)
		tmp = fma(beta, fma(beta, fma(beta, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.2], N[(beta * N[(beta * N[(beta * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{2}{81} \cdot \beta - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 70.8

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

   8. Applied rewrites 70.8%

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

   if 2.2000000000000002 < beta < 1.35000000000000003e154

   1. Initial program 89.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 77.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.5

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 92.5

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

   10. Applied rewrites 92.5%

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

Alternative 8: 97.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{2}{81} \cdot \beta - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 70.8

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

   8. Applied rewrites 70.8%

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

   if 2.2000000000000002 < beta

   1. Initial program 84.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-+.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 84.5

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

   7. Applied rewrites 84.5%

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

Alternative 9: 94.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{2}{81} \cdot \beta - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 70.8

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

   8. Applied rewrites 70.8%

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

   if 2.2000000000000002 < beta

   1. Initial program 84.4%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

Alternative 10: 91.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2) {
		tmp = fma(beta, fma(beta, fma(beta, 0.024691358024691357, -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 <= 2.2)
		tmp = fma(beta, fma(beta, fma(beta, 0.024691358024691357, -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, 2.2], N[(beta * N[(beta * N[(beta * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{2}{81} \cdot \beta - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. lower-fma.f64 70.8

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

   8. Applied rewrites 70.8%

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

   if 2.2000000000000002 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   8. Applied rewrites 78.4%

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

Alternative 11: 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 99.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{-5}{432} \cdot \beta - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-5}{432} \cdot \beta + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-5}{432}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \frac{-5}{432} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      6. lower-fma.f64 70.7

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

   8. Applied rewrites 70.7%

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

   if 1.6499999999999999 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   8. Applied rewrites 78.4%

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

Alternative 12: 47.8% 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 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in beta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 75.7

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

8. Applied rewrites 75.7%

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

9. Taylor expanded in alpha around 0

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

    1. lower-+.f64 73.5

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

11. Applied rewrites 73.5%

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

12. Taylor expanded in beta around 0

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

14. Applied rewrites 49.8%

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

15. Final simplification 49.8%

    \[\leadsto \frac{0.25}{\beta + 3} \]
16. Add Preprocessing

Alternative 13: 46.3% 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 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 49.7

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

8. Applied rewrites 49.7%

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

9. Final simplification 49.7%

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

Alternative 14: 45.6% 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 69.9

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

5. Applied rewrites 69.9%

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

6. Taylor expanded in beta around 0

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

8. Applied rewrites 48.7%

   \[\leadsto \color{blue}{0.08333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))