Octave 3.8, jcobi/3

Percentage Accurate: 94.7% → 99.6%

Time: 9.7s

Alternatives: 17

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\alpha \leq 85000000:\\ \;\;\;\;\frac{{t\_0}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)\right)}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(\left(\frac{-1}{\beta} - \alpha\right) + -1\right) - \frac{\alpha}{\beta}\right)}{\beta}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= alpha 85000000.0)
     (/
      (* (pow t_0 -2.0) (+ 1.0 (fma beta alpha (+ beta alpha))))
      (+ 3.0 (+ beta alpha)))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta))
        (- (+ (- (/ -1.0 beta) alpha) -1.0) (/ alpha beta)))
       beta)
      (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (alpha <= 85000000.0) {
		tmp = (pow(t_0, -2.0) * (1.0 + fma(beta, alpha, (beta + alpha)))) / (3.0 + (beta + alpha));
	} else {
		tmp = ((((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta)) - ((((-1.0 / beta) - alpha) + -1.0) - (alpha / beta))) / beta) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (alpha <= 85000000.0)
		tmp = Float64(Float64((t_0 ^ -2.0) * Float64(1.0 + fma(beta, alpha, Float64(beta + alpha)))) / Float64(3.0 + Float64(beta + alpha)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta)) - Float64(Float64(Float64(Float64(-1.0 / beta) - alpha) + -1.0) - Float64(alpha / beta))) / beta) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 85000000.0], N[(N[(N[Power[t$95$0, -2.0], $MachinePrecision] * N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-1.0 / beta), $MachinePrecision] - alpha), $MachinePrecision] + -1.0), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.5e7

   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

   4. Applied rewrites 99.9%

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

   if 8.5e7 < alpha

   1. Initial program 83.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 15.9

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

   5. Applied rewrites 15.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 85000000:\\ \;\;\;\;\frac{{\left(2 + \left(\beta + \alpha\right)\right)}^{-2} \cdot \left(1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)\right)}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(\left(\frac{-1}{\beta} - \alpha\right) + -1\right) - \frac{\alpha}{\beta}\right)}{\beta}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_0}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(\left(\frac{-1}{\beta} - \alpha\right) + -1\right) - \frac{\alpha}{\beta}\right)}{\beta}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1e+77)
     (/
      (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_0)
      (* (+ (+ 1.0 (+ beta alpha)) 2.0) t_0))
     (/
      (/
       (-
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta))
        (- (+ (- (/ -1.0 beta) alpha) -1.0) (/ alpha beta)))
       beta)
      (+ t_0 1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+77) {
		tmp = ((1.0 + fma(beta, alpha, (beta + alpha))) / t_0) / (((1.0 + (beta + alpha)) + 2.0) * t_0);
	} else {
		tmp = ((((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta)) - ((((-1.0 / beta) - alpha) + -1.0) - (alpha / beta))) / beta) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1e+77)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_0) / Float64(Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta)) - Float64(Float64(Float64(Float64(-1.0 / beta) - alpha) + -1.0) - Float64(alpha / beta))) / beta) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+77], N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-1.0 / beta), $MachinePrecision] - alpha), $MachinePrecision] + -1.0), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999983e76

   1. Initial program 99.7%

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

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 9.99999999999999983e76 < beta

   1. Initial program 81.2%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta} - \left(\left(\left(\frac{-1}{\beta} - \alpha\right) + -1\right) - \frac{\alpha}{\beta}\right)}{\beta}}{\left(2 + \left(\beta + \alpha\right)\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{t\_0}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1e+77)
     (/
      (/ (+ 1.0 (fma beta alpha (+ beta alpha))) t_0)
      (* (+ (+ 1.0 (+ beta alpha)) 2.0) t_0))
     (/ (/ (- alpha -1.0) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+77) {
		tmp = ((1.0 + fma(beta, alpha, (beta + alpha))) / t_0) / (((1.0 + (beta + alpha)) + 2.0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1e+77)
		tmp = Float64(Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / t_0) / Float64(Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+77], N[(N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.99999999999999983e76

   1. Initial program 99.7%

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

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 9.99999999999999983e76 < beta

   1. Initial program 81.2%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 84.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(t\_1 \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ 3.0 (+ beta alpha))))
   (if (<= beta 6e+100)
     (/ (+ 1.0 (fma beta alpha (+ beta alpha))) (* (* t_1 t_0) t_0))
     (/ (/ (- alpha -1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 6e+100) {
		tmp = (1.0 + fma(beta, alpha, (beta + alpha))) / ((t_1 * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_1) / t_0;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 6e+100)
		tmp = Float64(Float64(1.0 + fma(beta, alpha, Float64(beta + alpha))) / Float64(Float64(t_1 * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_1) / t_0);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6e+100], N[(N[(1.0 + N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.99999999999999971e100

   1. Initial program 99.7%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.0%

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

   if 5.99999999999999971e100 < beta

   1. Initial program 79.9%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 83.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta} - \frac{\frac{-1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+19)
   (/
    (/ (+ 1.0 beta) (+ 2.0 beta))
    (* (+ (+ 1.0 (+ beta alpha)) 2.0) (+ 2.0 (+ beta alpha))))
   (- (/ (/ alpha beta) beta) (/ (/ -1.0 beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+19) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * (2.0 + (beta + alpha)));
	} else {
		tmp = ((alpha / beta) / beta) - ((-1.0 / beta) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d+19) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (((1.0d0 + (beta + alpha)) + 2.0d0) * (2.0d0 + (beta + alpha)))
    else
        tmp = ((alpha / beta) / beta) - (((-1.0d0) / beta) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+19) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * (2.0 + (beta + alpha)));
	} else {
		tmp = ((alpha / beta) / beta) - ((-1.0 / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e+19:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * (2.0 + (beta + alpha)))
	else:
		tmp = ((alpha / beta) / beta) - ((-1.0 / beta) / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+19)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0) * Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha / beta) / beta) - Float64(Float64(-1.0 / beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e+19)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * (2.0 + (beta + alpha)));
	else
		tmp = ((alpha / beta) / beta) - ((-1.0 / beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.5e+19], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision] - N[(N[(-1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e19

   1. Initial program 99.7%

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

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 1.5e19 < beta

   1. Initial program 83.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 85.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta} - \frac{\frac{-1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1.6e+16)
     (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ (+ 1.0 (+ beta alpha)) 2.0) t_0))
     (/ (/ (- alpha -1.0) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.6e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 1.6d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (((1.0d0 + (beta + alpha)) + 2.0d0) * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / (3.0d0 + (beta + alpha))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.6e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.6e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * t_0)
	else:
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.6e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(Float64(1.0 + Float64(beta + alpha)) + 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.6e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (((1.0 + (beta + alpha)) + 2.0) * t_0);
	else
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.6e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 1.6e16 < beta

   1. Initial program 84.1%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 86.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(\left(1 + \left(\beta + \alpha\right)\right) + 2\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))) (t_1 (+ 3.0 (+ beta alpha))))
   (if (<= beta 1.45e+16)
     (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* t_1 t_0))
     (/ (/ (- alpha -1.0) t_1) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.45e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_1 * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_1) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    t_1 = 3.0d0 + (beta + alpha)
    if (beta <= 1.45d+16) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / (t_1 * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / t_1) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = 3.0 + (beta + alpha);
	double tmp;
	if (beta <= 1.45e+16) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_1 * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_1) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	t_1 = 3.0 + (beta + alpha)
	tmp = 0
	if beta <= 1.45e+16:
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_1 * t_0)
	else:
		tmp = ((alpha - -1.0) / t_1) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(3.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.45e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_1) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	t_1 = 3.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1.45e+16)
		tmp = ((1.0 + beta) / (2.0 + beta)) / (t_1 * t_0);
	else
		tmp = ((alpha - -1.0) / t_1) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.45e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45e16

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if 1.45e16 < beta

   1. Initial program 84.1%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.0

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

   5. Applied rewrites 86.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 86.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 11.6:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \alpha}}{\left(3 + \alpha\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= beta 11.6)
     (/ (/ (- alpha -1.0) (+ 2.0 alpha)) (* (+ 3.0 alpha) t_0))
     (/ (/ (- alpha -1.0) (+ 3.0 (+ beta alpha))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 11.6) {
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((3.0 + alpha) * t_0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (beta + alpha)
    if (beta <= 11.6d0) then
        tmp = ((alpha - (-1.0d0)) / (2.0d0 + alpha)) / ((3.0d0 + alpha) * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / (3.0d0 + (beta + alpha))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 11.6) {
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((3.0 + alpha) * t_0);
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 11.6:
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((3.0 + alpha) * t_0)
	else:
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 11.6)
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(2.0 + alpha)) / Float64(Float64(3.0 + alpha) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(beta + alpha))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 11.6)
		tmp = ((alpha - -1.0) / (2.0 + alpha)) / ((3.0 + alpha) * t_0);
	else
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 11.6], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 11.5999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in beta around 0

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

      1. lower-+.f64 97.4

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

   10. Applied rewrites 97.4%

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

   if 11.5999999999999996 < beta

   1. Initial program 85.5%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 82.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11.6:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \alpha}}{\left(3 + \alpha\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.82:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.82)
   (/ (fma (fma -0.125 alpha 0.25) alpha 0.5) (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (/ (/ (- alpha -1.0) (+ 3.0 (+ beta alpha))) (+ 2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.82) {
		tmp = fma(fma(-0.125, alpha, 0.25), alpha, 0.5) / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((alpha - -1.0) / (3.0 + (beta + alpha))) / (2.0 + (beta + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.82)
		tmp = Float64(fma(fma(-0.125, alpha, 0.25), alpha, 0.5) / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / Float64(3.0 + Float64(beta + alpha))) / Float64(2.0 + Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 0.82], N[(N[(N[(-0.125 * alpha + 0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.819999999999999951

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{2} + \color{blue}{\alpha \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \color{blue}{\alpha}, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 69.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 69.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 0.819999999999999951 < beta

   1. Initial program 85.5%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

   7. Applied rewrites 82.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.82:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{3 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.0)
   (/ (fma (fma -0.125 alpha 0.25) alpha 0.5) (* (+ 3.0 alpha) (+ 2.0 alpha)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.0) {
		tmp = fma(fma(-0.125, alpha, 0.25), alpha, 0.5) / ((3.0 + alpha) * (2.0 + alpha));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.0)
		tmp = Float64(fma(fma(-0.125, alpha, 0.25), alpha, 0.5) / Float64(Float64(3.0 + alpha) * Float64(2.0 + alpha)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.0], N[(N[(N[(-0.125 * alpha + 0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision] / N[(N[(3.0 + alpha), $MachinePrecision] * N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1}{2} + \color{blue}{\alpha \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \alpha\right)}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \color{blue}{\alpha}, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]

   11. Taylor expanded in beta around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, \alpha, \frac{1}{4}\right), \alpha, \frac{1}{2}\right)}{\color{blue}{\left(3 + \alpha\right)} \cdot \left(2 + \alpha\right)} \]

       4. lower-+.f64 69.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \color{blue}{\left(2 + \alpha\right)}} \]

   13. Applied rewrites 69.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\color{blue}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}} \]

   if 3 < beta

   1. Initial program 85.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \alpha, 0.25\right), \alpha, 0.5\right)}{\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 52:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 52.0)
   (/ (fma 0.25 alpha 0.5) (* (+ 3.0 (+ beta alpha)) (+ 2.0 (+ beta alpha))))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 52.0) {
		tmp = fma(0.25, alpha, 0.5) / ((3.0 + (beta + alpha)) * (2.0 + (beta + alpha)));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 52.0)
		tmp = Float64(fma(0.25, alpha, 0.5) / Float64(Float64(3.0 + Float64(beta + alpha)) * Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 52.0], N[(N[(0.25 * alpha + 0.5), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 52

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 83.4%

       \[\leadsto \frac{\mathsf{fma}\left(0.25, \color{blue}{\alpha}, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]

   if 52 < beta

   1. Initial program 85.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 52:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, 0.5\right)}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{elif}\;\beta \leq 10^{+160}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.5 (* (+ 3.0 beta) (+ 2.0 beta)))
   (if (<= beta 1e+160)
     (/ (- alpha -1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	} else if (beta <= 1e+160) {
		tmp = (alpha - -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 (beta <= 9.0d0) then
        tmp = 0.5d0 / ((3.0d0 + beta) * (2.0d0 + beta))
    else if (beta <= 1d+160) then
        tmp = (alpha - (-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 (beta <= 9.0) {
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	} else if (beta <= 1e+160) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta))
	elif beta <= 1e+160:
		tmp = (alpha - -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 (beta <= 9.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + beta) * Float64(2.0 + beta)));
	elseif (beta <= 1e+160)
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	elseif (beta <= 1e+160)
		tmp = (alpha - -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[beta, 9.0], N[(0.5 / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1e+160], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 9

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 84.4%

       \[\leadsto \frac{0.5}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]

   11. Taylor expanded in alpha around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 69.7

          \[\leadsto \frac{0.5}{\left(2 + \beta\right) \cdot \color{blue}{\left(3 + \beta\right)}} \]

   13. Applied rewrites 69.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 9 < beta < 1.00000000000000001e160

   1. Initial program 91.2%

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

   3. Taylor expanded in 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 76.1

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

   5. Applied rewrites 76.1%

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

   if 1.00000000000000001e160 < beta

   1. Initial program 78.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 87.3%

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

   10. Applied rewrites 86.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{elif}\;\beta \leq 10^{+160}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (/ 0.5 (* (+ 3.0 beta) (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.0d0) then
        tmp = 0.5d0 / ((3.0d0 + beta) * (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9.0:
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = Float64(0.5 / Float64(Float64(3.0 + beta) * Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.0)
		tmp = 0.5 / ((3.0 + beta) * (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.0], N[(0.5 / N[(N[(3.0 + beta), $MachinePrecision] * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 84.4%

       \[\leadsto \frac{0.5}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)} \]

   11. Taylor expanded in alpha around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 69.7

          \[\leadsto \frac{0.5}{\left(2 + \beta\right) \cdot \color{blue}{\left(3 + \beta\right)}} \]

   13. Applied rewrites 69.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]

   if 9 < beta

   1. Initial program 85.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{0.5}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+160}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+160)
   (/ (- alpha -1.0) (* beta beta))
   (/ (/ alpha beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+160) {
		tmp = (alpha - -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 (beta <= 1d+160) then
        tmp = (alpha - (-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 (beta <= 1e+160) {
		tmp = (alpha - -1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1e+160:
		tmp = (alpha - -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 (beta <= 1e+160)
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+160)
		tmp = (alpha - -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[beta, 1e+160], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000001e160

   1. Initial program 97.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 19.1

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

   5. Applied rewrites 19.1%

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

   if 1.00000000000000001e160 < beta

   1. Initial program 78.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 87.3%

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

   10. Applied rewrites 86.0%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+160}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], 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 < 1

   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 33.4

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

   5. Applied rewrites 33.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 33.4%

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

   if 1 < alpha

   1. Initial program 83.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 17.2

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 17.2%

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

Alternative 16: 53.3% accurate, 4.2× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ (- alpha -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(alpha - -1.0) / Float64(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[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in 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 28.7

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

5. Applied rewrites 28.7%

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

6. Final simplification 28.7%

   \[\leadsto \frac{\alpha - -1}{\beta \cdot \beta} \]
7. Add Preprocessing

Alternative 17: 32.0% 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 95.2%

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

3. Taylor expanded in 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 28.7

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

5. Applied rewrites 28.7%

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

6. Taylor expanded in alpha around inf

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

8. Applied rewrites 16.8%

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

Reproduce

herbie shell --seed 2024285 
(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)))