Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.6%

Time: 8.5s

Alternatives: 17

Speedup: 1.9×

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: 94.6% 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, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \frac{1 + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(t\_1 + \alpha\right) + 1\right) \cdot \beta}{t\_0}}{t\_0}}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\sqrt{\beta}, \sqrt{\beta}, 3 + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (/ (+ 1.0 alpha) beta)))
   (if (<= beta 4e+149)
     (/ (/ (/ (* (+ (+ t_1 alpha) 1.0) beta) t_0) t_0) (+ t_0 1.0))
     (/ t_1 (fma (sqrt beta) (sqrt beta) (+ 3.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (1.0 + alpha) / beta;
	double tmp;
	if (beta <= 4e+149) {
		tmp = (((((t_1 + alpha) + 1.0) * beta) / t_0) / t_0) / (t_0 + 1.0);
	} else {
		tmp = t_1 / fma(sqrt(beta), sqrt(beta), (3.0 + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(1.0 + alpha) / beta)
	tmp = 0.0
	if (beta <= 4e+149)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(t_1 + alpha) + 1.0) * beta) / t_0) / t_0) / Float64(t_0 + 1.0));
	else
		tmp = Float64(t_1 / fma(sqrt(beta), sqrt(beta), Float64(3.0 + alpha)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]}, If[LessEqual[beta, 4e+149], N[(N[(N[(N[(N[(N[(t$95$1 + alpha), $MachinePrecision] + 1.0), $MachinePrecision] * beta), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[Sqrt[beta], $MachinePrecision] * N[Sqrt[beta], $MachinePrecision] + N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000002e149

   1. Initial program 97.5%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if 4.0000000000000002e149 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. unpow1 N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow1 N/A

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

      15. pow2 N/A

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

      16. sqrt-prod N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 92.1

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

   7. Applied rewrites 92.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\frac{1 + \alpha}{\beta} + \alpha\right) + 1\right) \cdot \beta}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\mathsf{fma}\left(\sqrt{\beta}, \sqrt{\beta}, 3 + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\mathsf{fma}\left(\sqrt{\beta}, \sqrt{\beta}, 3 + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 4e+149)
     (/
      (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)
      (+ (+ 1.0 (+ beta alpha)) 2.0))
     (/ (/ (+ 1.0 alpha) beta) (fma (sqrt beta) (sqrt beta) (+ 3.0 alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 4e+149) {
		tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / ((1.0 + (beta + alpha)) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / fma(sqrt(beta), sqrt(beta), (3.0 + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 4e+149)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / fma(sqrt(beta), sqrt(beta), Float64(3.0 + alpha)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e+149], 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[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[Sqrt[beta], $MachinePrecision] * N[Sqrt[beta], $MachinePrecision] + N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.0000000000000002e149

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 97.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 97.6

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 97.6

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

   4. Applied rewrites 97.6%

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

   if 4.0000000000000002e149 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. unpow1 N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow1 N/A

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

      15. pow2 N/A

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

      16. sqrt-prod N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 92.1

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

   7. Applied rewrites 92.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\mathsf{fma}\left(\sqrt{\beta}, \sqrt{\beta}, 3 + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{t\_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+49)
     (/
      (/ (+ (fma beta alpha (+ beta alpha)) 1.0) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+49) {
		tmp = ((fma(beta, alpha, (beta + alpha)) + 1.0) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+49)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+49], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000004e49

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 98.3%

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

   if 5.0000000000000004e49 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 86.4%

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

Alternative 4: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{2 + \beta}}{\left(2 + \alpha\right) + \beta}}{3 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.1e-14)
   (/
    (+ (fma beta alpha (+ beta alpha)) 1.0)
    (* (+ (+ beta alpha) 2.0) (* (+ 2.0 alpha) (+ 3.0 alpha))))
   (if (<= beta 2.6e+15)
     (/ (/ (/ (+ beta 1.0) (+ 2.0 beta)) (+ (+ 2.0 alpha) beta)) (+ 3.0 beta))
     (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e-14) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (((beta + alpha) + 2.0) * ((2.0 + alpha) * (3.0 + alpha)));
	} else if (beta <= 2.6e+15) {
		tmp = (((beta + 1.0) / (2.0 + beta)) / ((2.0 + alpha) + beta)) / (3.0 + beta);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.1e-14)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(Float64(beta + alpha) + 2.0) * Float64(Float64(2.0 + alpha) * Float64(3.0 + alpha))));
	elseif (beta <= 2.6e+15)
		tmp = Float64(Float64(Float64(Float64(beta + 1.0) / Float64(2.0 + beta)) / Float64(Float64(2.0 + alpha) + beta)) / Float64(3.0 + beta));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.1e-14], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.6e+15], N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.1000000000000002e-14

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 92.1

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

   7. Applied rewrites 92.1%

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

   if 4.1000000000000002e-14 < beta < 2.6e15

   1. Initial program 99.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. associate--l+ N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. lower-+.f64 N/A

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

   7. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

   9. Applied rewrites 91.4%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-+.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if 2.6e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2e-14)
   (/
    (+ (fma beta alpha (+ beta alpha)) 1.0)
    (* (+ (+ beta alpha) 2.0) (* (+ 2.0 alpha) (+ 3.0 alpha))))
   (if (<= beta 9e+15)
     (/
      (/ (+ beta 1.0) (+ 2.0 beta))
      (* (+ (+ 3.0 alpha) beta) (+ (+ 2.0 alpha) beta)))
     (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2e-14) {
		tmp = (fma(beta, alpha, (beta + alpha)) + 1.0) / (((beta + alpha) + 2.0) * ((2.0 + alpha) * (3.0 + alpha)));
	} else if (beta <= 9e+15) {
		tmp = ((beta + 1.0) / (2.0 + beta)) / (((3.0 + alpha) + beta) * ((2.0 + alpha) + beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2e-14)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) + 1.0) / Float64(Float64(Float64(beta + alpha) + 2.0) * Float64(Float64(2.0 + alpha) * Float64(3.0 + alpha))));
	elseif (beta <= 9e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(2.0 + beta)) / Float64(Float64(Float64(3.0 + alpha) + beta) * Float64(Float64(2.0 + alpha) + beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.2e-14], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.2000000000000002e-14

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 92.1

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

   7. Applied rewrites 92.1%

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

   if 3.2000000000000002e-14 < beta < 9e15

   1. Initial program 99.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 91.0%

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

   if 9e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

Alternative 6: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}}{t\_0}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{2 + \beta}}{\left(\left(3 + \alpha\right) + \beta\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 alpha) beta)))
   (if (<= beta 4.5e-14)
     (/ (/ (+ 1.0 alpha) (* (+ 3.0 alpha) (+ 2.0 alpha))) t_0)
     (if (<= beta 9e+15)
       (/ (/ (+ beta 1.0) (+ 2.0 beta)) (* (+ (+ 3.0 alpha) beta) t_0))
       (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	} else if (beta <= 9e+15) {
		tmp = ((beta + 1.0) / (2.0 + beta)) / (((3.0 + alpha) + beta) * t_0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + alpha) + beta
    if (beta <= 4.5d-14) then
        tmp = ((1.0d0 + alpha) / ((3.0d0 + alpha) * (2.0d0 + alpha))) / t_0
    else if (beta <= 9d+15) then
        tmp = ((beta + 1.0d0) / (2.0d0 + beta)) / (((3.0d0 + alpha) + beta) * t_0)
    else
        tmp = ((1.0d0 + alpha) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	} else if (beta <= 9e+15) {
		tmp = ((beta + 1.0) / (2.0 + beta)) / (((3.0 + alpha) + beta) * t_0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (2.0 + alpha) + beta
	tmp = 0
	if beta <= 4.5e-14:
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0
	elif beta <= 9e+15:
		tmp = ((beta + 1.0) / (2.0 + beta)) / (((3.0 + alpha) + beta) * t_0)
	else:
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 4.5e-14)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha))) / t_0);
	elseif (beta <= 9e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(2.0 + beta)) / Float64(Float64(Float64(3.0 + alpha) + beta) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (2.0 + alpha) + beta;
	tmp = 0.0;
	if (beta <= 4.5e-14)
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	elseif (beta <= 9e+15)
		tmp = ((beta + 1.0) / (2.0 + beta)) / (((3.0 + alpha) + beta) * t_0);
	else
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 4.5e-14], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[beta, 9e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + alpha), $MachinePrecision] + beta), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.4999999999999998e-14

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 97.2

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

   9. Applied rewrites 97.2%

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

   if 4.4999999999999998e-14 < beta < 9e15

   1. Initial program 99.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 91.0%

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

   if 9e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

Alternative 7: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{t\_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 7e+49)
     (/ (* (+ 1.0 alpha) (+ 1.0 beta)) (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
     (/ (/ (+ alpha 1.0) beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 7e+49) {
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 7d+49) then
        tmp = ((1.0d0 + alpha) * (1.0d0 + beta)) / (t_0 * ((3.0d0 + (beta + alpha)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 7e+49) {
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 7e+49:
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_0 * ((3.0 + (beta + alpha)) * t_0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 7e+49)
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + beta)) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 7e+49)
		tmp = ((1.0 + alpha) * (1.0 + beta)) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 7e+49], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.9999999999999995e49

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in alpha around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if 6.9999999999999995e49 < beta

   1. Initial program 86.1%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 86.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(2 + \alpha\right) + \beta\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}}{t\_0}\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ 2.0 alpha) beta)))
   (if (<= beta 4.5e-14)
     (/ (/ (+ 1.0 alpha) (* (+ 3.0 alpha) (+ 2.0 alpha))) t_0)
     (if (<= beta 2.6e+15)
       (/ (/ (+ 1.0 beta) (* (+ 3.0 beta) (+ 2.0 beta))) t_0)
       (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	} else if (beta <= 2.6e+15) {
		tmp = ((1.0 + beta) / ((3.0 + beta) * (2.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + alpha) + beta
    if (beta <= 4.5d-14) then
        tmp = ((1.0d0 + alpha) / ((3.0d0 + alpha) * (2.0d0 + alpha))) / t_0
    else if (beta <= 2.6d+15) then
        tmp = ((1.0d0 + beta) / ((3.0d0 + beta) * (2.0d0 + beta))) / t_0
    else
        tmp = ((1.0d0 + alpha) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (2.0 + alpha) + beta;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	} else if (beta <= 2.6e+15) {
		tmp = ((1.0 + beta) / ((3.0 + beta) * (2.0 + beta))) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (2.0 + alpha) + beta
	tmp = 0
	if beta <= 4.5e-14:
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0
	elif beta <= 2.6e+15:
		tmp = ((1.0 + beta) / ((3.0 + beta) * (2.0 + beta))) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(2.0 + alpha) + beta)
	tmp = 0.0
	if (beta <= 4.5e-14)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha))) / t_0);
	elseif (beta <= 2.6e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(3.0 + beta) * Float64(2.0 + beta))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (2.0 + alpha) + beta;
	tmp = 0.0;
	if (beta <= 4.5e-14)
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / t_0;
	elseif (beta <= 2.6e+15)
		tmp = ((1.0 + beta) / ((3.0 + beta) * (2.0 + beta))) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[beta, 4.5e-14], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[beta, 2.6e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.4999999999999998e-14

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 97.2

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

   9. Applied rewrites 97.2%

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

   if 4.4999999999999998e-14 < beta < 2.6e15

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

   6. Applied rewrites 97.4%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 72.5

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

   9. Applied rewrites 72.5%

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

   if 2.6e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

Alternative 9: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}}{\left(2 + \alpha\right) + \beta}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 4.5e-14)
     (/
      (/ (+ 1.0 alpha) (* (+ 3.0 alpha) (+ 2.0 alpha)))
      (+ (+ 2.0 alpha) beta))
     (if (<= beta 9e+15)
       (/ (+ 1.0 beta) (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
       (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / ((2.0 + alpha) + beta);
	} else if (beta <= 9e+15) {
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 4.5d-14) then
        tmp = ((1.0d0 + alpha) / ((3.0d0 + alpha) * (2.0d0 + alpha))) / ((2.0d0 + alpha) + beta)
    else if (beta <= 9d+15) then
        tmp = (1.0d0 + beta) / (t_0 * ((3.0d0 + (beta + alpha)) * t_0))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 4.5e-14) {
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / ((2.0 + alpha) + beta);
	} else if (beta <= 9e+15) {
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 4.5e-14:
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / ((2.0 + alpha) + beta)
	elif beta <= 9e+15:
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0))
	else:
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 4.5e-14)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha))) / Float64(Float64(2.0 + alpha) + beta));
	elseif (beta <= 9e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 4.5e-14)
		tmp = ((1.0 + alpha) / ((3.0 + alpha) * (2.0 + alpha))) / ((2.0 + alpha) + beta);
	elseif (beta <= 9e+15)
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	else
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 4.5e-14], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.4999999999999998e-14

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.9%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 97.2

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

   9. Applied rewrites 97.2%

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

   if 4.4999999999999998e-14 < beta < 9e15

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 9e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}}{\left(2 + \alpha\right) + \beta}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t\_0 \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

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 9e+15)
     (/ (+ 1.0 beta) (* t_0 (* (+ 3.0 (+ beta alpha)) t_0)))
     (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 9e+15) {
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 9d+15) then
        tmp = (1.0d0 + beta) / (t_0 * ((3.0d0 + (beta + alpha)) * t_0))
    else
        tmp = ((1.0d0 + alpha) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 9e+15) {
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 9e+15:
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0))
	else:
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 9e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(Float64(3.0 + Float64(beta + alpha)) * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 9e+15)
		tmp = (1.0 + beta) / (t_0 * ((3.0 + (beta + alpha)) * t_0));
	else
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e15

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 81.2

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

   7. Applied rewrites 81.2%

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

   if 9e15 < beta

   1. Initial program 85.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.1:\\ \;\;\;\;\frac{\frac{0.5}{\left(2 + \alpha\right) + \beta}}{\left(\left(\beta + \alpha\right) + 1\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\beta + \alpha\right) + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.1)
   (/ (/ 0.5 (+ (+ 2.0 alpha) beta)) (+ (+ (+ beta alpha) 1.0) 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ beta alpha) 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.1) {
		tmp = (0.5 / ((2.0 + alpha) + beta)) / (((beta + alpha) + 1.0) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.1d0) then
        tmp = (0.5d0 / ((2.0d0 + alpha) + beta)) / (((beta + alpha) + 1.0d0) + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / ((beta + alpha) + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.1) {
		tmp = (0.5 / ((2.0 + alpha) + beta)) / (((beta + alpha) + 1.0) + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.1:
		tmp = (0.5 / ((2.0 + alpha) + beta)) / (((beta + alpha) + 1.0) + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.1)
		tmp = Float64(Float64(0.5 / Float64(Float64(2.0 + alpha) + beta)) / Float64(Float64(Float64(beta + alpha) + 1.0) + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(beta + alpha) + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.1)
		tmp = (0.5 / ((2.0 + alpha) + beta)) / (((beta + alpha) + 1.0) + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / ((beta + alpha) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.1], N[(N[(0.5 / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.0999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. associate--l+ N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. lower-+.f64 N/A

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

   7. Applied rewrites 83.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

   9. Applied rewrites 83.2%

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

   10. Taylor expanded in beta around 0

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

   12. Applied rewrites 79.9%

       \[\leadsto \frac{\frac{0.5}{\left(2 + \alpha\right) + \beta}}{\left(\left(\beta + \alpha\right) + 1\right) + 2} \]

   if 7.0999999999999996 < beta

   1. Initial program 86.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 79.7

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

   7. Applied rewrites 79.7%

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

Alternative 12: 55.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.4:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.4) (/ (/ 1.0 beta) beta) (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.4) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.4d0) then
        tmp = (1.0d0 / beta) / beta
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.4) {
		tmp = (1.0 / beta) / beta;
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 0.4:
		tmp = (1.0 / beta) / beta
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.4)
		tmp = Float64(Float64(1.0 / beta) / beta);
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.4)
		tmp = (1.0 / beta) / beta;
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 0.4], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.40000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 35.3%

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

   10. Applied rewrites 35.9%

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

   if 0.40000000000000002 < alpha

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 13.2%

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

   10. Applied rewrites 12.9%

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

Alternative 13: 55.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5e+42) (/ (+ 1.0 alpha) (* beta beta)) (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e+42) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5d+42) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e+42) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 5e+42:
		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 (alpha <= 5e+42)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5e+42)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 5e+42], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.00000000000000007e42

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 34.3

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

   5. Applied rewrites 34.3%

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

   if 5.00000000000000007e42 < alpha

   1. Initial program 85.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 11.5%

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

   10. Applied rewrites 11.2%

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

Alternative 14: 55.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{\frac{\alpha + 1}{\beta}}{\beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) / beta) / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return ((alpha + 1.0) / beta) / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return ((alpha + 1.0) / beta) / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(Float64(alpha + 1.0) / beta) / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) / beta) / beta;
end

Wolfram

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

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
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 27.4

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

5. Applied rewrites 27.4%

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

7. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\beta}}{\beta}} \]
8. Add Preprocessing

Alternative 15: 52.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.7e-11) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.7e-11) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.7d-11) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.7e-11) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 2.7e-11:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.7e-11)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.7e-11)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 2.7e-11], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.70000000000000005e-11

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 35.1%

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

   if 2.70000000000000005e-11 < alpha

   1. Initial program 88.3%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   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 14.2

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 14.0%

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

Alternative 16: 52.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1 + \alpha}{\beta \cdot \beta} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Fortran

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

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return (1.0 + alpha) / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return (1.0 + alpha) / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(Float64(1.0 + alpha) / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) / (beta * beta);
end

Wolfram

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

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
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 27.4

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

5. Applied rewrites 27.4%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
6. Add Preprocessing

Alternative 17: 31.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
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 27.4

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

5. Applied rewrites 27.4%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 15.6%

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

Reproduce

herbie shell --seed 2024327 
(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)))