Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.6%

Time: 14.0s

Alternatives: 22

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% 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, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\mathsf{fma}\left(\beta + \alpha, \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right), 8\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{\mathsf{fma}\left(\beta + \alpha, \beta + \alpha, 4\right) - \left(\beta + \alpha\right) \cdot 2}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \frac{1}{\beta}\right) + \left(1 + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8e+90)
   (*
    (/
     (/
      (+ alpha (+ beta (fma alpha beta 1.0)))
      (fma (+ beta alpha) (* (+ beta alpha) (+ beta alpha)) 8.0))
     (+ alpha (+ beta 3.0)))
    (/
     (- (fma (+ beta alpha) (+ beta alpha) 4.0) (* (+ beta alpha) 2.0))
     (+ alpha (+ beta 2.0))))
   (/
    (/
     (+
      (+ (+ alpha (/ 1.0 beta)) (+ 1.0 (/ alpha beta)))
      (* (- -1.0 alpha) (/ (+ alpha 2.0) beta)))
     (+ (+ beta alpha) 2.0))
    (+ 2.0 (+ 1.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8e+90) {
		tmp = (((alpha + (beta + fma(alpha, beta, 1.0))) / fma((beta + alpha), ((beta + alpha) * (beta + alpha)), 8.0)) / (alpha + (beta + 3.0))) * ((fma((beta + alpha), (beta + alpha), 4.0) - ((beta + alpha) * 2.0)) / (alpha + (beta + 2.0)));
	} else {
		tmp = ((((alpha + (1.0 / beta)) + (1.0 + (alpha / beta))) + ((-1.0 - alpha) * ((alpha + 2.0) / beta))) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8e+90)
		tmp = Float64(Float64(Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / fma(Float64(beta + alpha), Float64(Float64(beta + alpha) * Float64(beta + alpha)), 8.0)) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(fma(Float64(beta + alpha), Float64(beta + alpha), 4.0) - Float64(Float64(beta + alpha) * 2.0)) / Float64(alpha + Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + Float64(1.0 / beta)) + Float64(1.0 + Float64(alpha / beta))) + Float64(Float64(-1.0 - alpha) * Float64(Float64(alpha + 2.0) / beta))) / Float64(Float64(beta + alpha) + 2.0)) / Float64(2.0 + Float64(1.0 + 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, 2.8e+90], N[(N[(N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(beta + alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision] + 4.0), $MachinePrecision] - N[(N[(beta + alpha), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e90

   1. Initial program 99.3%

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

      1. associate-/l/ N/A

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

      2. metadata-eval N/A

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

      3. flip3-+ N/A

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

      4. associate-/r/ N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 79.4%

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

   if 2.8e90 < beta

   1. Initial program 84.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. *-lowering-*.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-lowering-+.f64 92.3

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

   7. Simplified 92.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\mathsf{fma}\left(\beta + \alpha, \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right), 8\right)}}{\alpha + \left(\beta + 3\right)} \cdot \frac{\mathsf{fma}\left(\beta + \alpha, \beta + \alpha, 4\right) - \left(\beta + \alpha\right) \cdot 2}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \frac{1}{\beta}\right) + \left(1 + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \frac{1}{\beta}\right) + \left(1 + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4.5e+93)
     (/
      (+ alpha (+ beta (fma alpha beta 1.0)))
      (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (/
      (/
       (+
        (+ (+ alpha (/ 1.0 beta)) (+ 1.0 (/ alpha beta)))
        (* (- -1.0 alpha) (/ (+ alpha 2.0) beta)))
       (+ (+ beta alpha) 2.0))
      (+ 2.0 (+ 1.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4.5e+93) {
		tmp = (alpha + (beta + fma(alpha, beta, 1.0))) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((((alpha + (1.0 / beta)) + (1.0 + (alpha / beta))) + ((-1.0 - alpha) * ((alpha + 2.0) / beta))) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 4.5e+93)
		tmp = Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + Float64(1.0 / beta)) + Float64(1.0 + Float64(alpha / beta))) + Float64(Float64(-1.0 - alpha) * Float64(Float64(alpha + 2.0) / beta))) / Float64(Float64(beta + alpha) + 2.0)) / Float64(2.0 + Float64(1.0 + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.5e+93], N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.49999999999999991e93

   1. Initial program 99.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 94.5%

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

   if 4.49999999999999991e93 < beta

   1. Initial program 84.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. *-lowering-*.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-lowering-+.f64 92.3

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

   7. Simplified 92.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \frac{1}{\beta}\right) + \left(1 + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+91}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}\right)}{\beta}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+91)
     (/
      (+ alpha (+ beta (fma alpha beta 1.0)))
      (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (/
      (/
       (+
        (+ (/ 1.0 beta) (+ alpha (/ alpha beta)))
        (+ 1.0 (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta))))
       beta)
      (+ 2.0 (+ 1.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+91) {
		tmp = (alpha + (beta + fma(alpha, beta, 1.0))) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta)))) / beta) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+91)
		tmp = Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha + Float64(alpha / beta))) + Float64(1.0 + Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta)))) / beta) / Float64(2.0 + Float64(1.0 + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+91], N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000008e91

   1. Initial program 99.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 94.5%

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

   if 1.00000000000000008e91 < beta

   1. Initial program 84.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 92.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+91}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}\right)}{\beta}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{t\_0}}{t\_0}}{1 + t\_0} \leq 2.4 \cdot 10^{-150}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<=
        (/
         (/ (/ (+ 1.0 (+ (+ beta alpha) (* beta alpha))) t_0) t_0)
         (+ 1.0 t_0))
        2.4e-150)
     (* (* alpha alpha) 0.0625)
     (/ 0.25 (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2.4e-150) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.25 / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (((((1.0d0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0d0 + t_0)) <= 2.4d-150) then
        tmp = (alpha * alpha) * 0.0625d0
    else
        tmp = 0.25d0 / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2.4e-150) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.25 / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2.4e-150:
		tmp = (alpha * alpha) * 0.0625
	else:
		tmp = 0.25 / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + Float64(Float64(beta + alpha) + Float64(beta * alpha))) / t_0) / t_0) / Float64(1.0 + t_0)) <= 2.4e-150)
		tmp = Float64(Float64(alpha * alpha) * 0.0625);
	else
		tmp = Float64(0.25 / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2.4e-150)
		tmp = (alpha * alpha) * 0.0625;
	else
		tmp = 0.25 / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 2.4e-150], N[(N[(alpha * alpha), $MachinePrecision] * 0.0625), $MachinePrecision], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 2.4e-150

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} + {\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right) + \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\frac{1}{16} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\alpha \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \alpha \cdot \frac{1}{16} + \color{blue}{\frac{-1}{16}}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. accelerator-lowering-fma.f64 4.2

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.0625, -0.0625\right)}, 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Simplified 4.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot {\alpha}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \frac{1}{16} \]

       4. *-lowering-*.f64 18.5

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot 0.0625 \]

   11. Simplified 18.5%

       \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot 0.0625} \]

   if 2.4e-150 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 93.4%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 83.7

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

   5. Simplified 83.7%

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

   6. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 70.4

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

   8. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 2.4 \cdot 10^{-150}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{t\_0}}{t\_0}}{1 + t\_0} \leq 2 \cdot 10^{-107}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<=
        (/
         (/ (/ (+ 1.0 (+ (+ beta alpha) (* beta alpha))) t_0) t_0)
         (+ 1.0 t_0))
        2e-107)
     (* (* alpha alpha) 0.0625)
     (/ 0.25 (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-107) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.25 / (alpha + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (((((1.0d0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0d0 + t_0)) <= 2d-107) then
        tmp = (alpha * alpha) * 0.0625d0
    else
        tmp = 0.25d0 / (alpha + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-107) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.25 / (alpha + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-107:
		tmp = (alpha * alpha) * 0.0625
	else:
		tmp = 0.25 / (alpha + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + Float64(Float64(beta + alpha) + Float64(beta * alpha))) / t_0) / t_0) / Float64(1.0 + t_0)) <= 2e-107)
		tmp = Float64(Float64(alpha * alpha) * 0.0625);
	else
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-107)
		tmp = (alpha * alpha) * 0.0625;
	else
		tmp = 0.25 / (alpha + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 2e-107], N[(N[(alpha * alpha), $MachinePrecision] * 0.0625), $MachinePrecision], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 2e-107

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} + {\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right) + \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\frac{1}{16} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\alpha \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \alpha \cdot \frac{1}{16} + \color{blue}{\frac{-1}{16}}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. accelerator-lowering-fma.f64 4.3

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.0625, -0.0625\right)}, 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Simplified 4.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot {\alpha}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \frac{1}{16} \]

       4. *-lowering-*.f64 17.1

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot 0.0625 \]

   11. Simplified 17.1%

       \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot 0.0625} \]

   if 2e-107 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 92.9%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 76.2%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 75.2

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

   11. Simplified 75.2%

       \[\leadsto \frac{0.25}{\color{blue}{\alpha + 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 2 \cdot 10^{-107}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{t\_0}}{t\_0}}{1 + t\_0} \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<=
        (/
         (/ (/ (+ 1.0 (+ (+ beta alpha) (* beta alpha))) t_0) t_0)
         (+ 1.0 t_0))
        2e-103)
     (* (* alpha alpha) 0.0625)
     0.08333333333333333)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-103) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.08333333333333333;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (((((1.0d0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0d0 + t_0)) <= 2d-103) then
        tmp = (alpha * alpha) * 0.0625d0
    else
        tmp = 0.08333333333333333d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-103) {
		tmp = (alpha * alpha) * 0.0625;
	} else {
		tmp = 0.08333333333333333;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-103:
		tmp = (alpha * alpha) * 0.0625
	else:
		tmp = 0.08333333333333333
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + Float64(Float64(beta + alpha) + Float64(beta * alpha))) / t_0) / t_0) / Float64(1.0 + t_0)) <= 2e-103)
		tmp = Float64(Float64(alpha * alpha) * 0.0625);
	else
		tmp = 0.08333333333333333;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((((1.0 + ((beta + alpha) + (beta * alpha))) / t_0) / t_0) / (1.0 + t_0)) <= 2e-103)
		tmp = (alpha * alpha) * 0.0625;
	else
		tmp = 0.08333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 2e-103], N[(N[(alpha * alpha), $MachinePrecision] * 0.0625), $MachinePrecision], 0.08333333333333333]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 1.99999999999999992e-103

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 48.0

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

   5. Simplified 48.0%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} + {\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right) + \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\frac{1}{16} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\alpha \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \alpha \cdot \frac{1}{16} + \color{blue}{\frac{-1}{16}}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. accelerator-lowering-fma.f64 4.3

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.0625, -0.0625\right)}, 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Simplified 4.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot {\alpha}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \frac{1}{16} \]

       4. *-lowering-*.f64 16.8

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot 0.0625 \]

   11. Simplified 16.8%

       \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot 0.0625} \]

   if 1.99999999999999992e-103 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 92.8%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 76.8

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

   8. Simplified 76.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 75.5%

       \[\leadsto \color{blue}{0.08333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+93)
     (/
      (+ alpha (+ beta (fma alpha beta 1.0)))
      (* (+ alpha (+ beta 3.0)) (* t_0 t_0)))
     (/
      (/ (+ alpha 1.0) (+ (+ beta alpha) 2.0))
      (+ 2.0 (+ 1.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+93) {
		tmp = (alpha + (beta + fma(alpha, beta, 1.0))) / ((alpha + (beta + 3.0)) * (t_0 * t_0));
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+93)
		tmp = Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / Float64(Float64(alpha + Float64(beta + 3.0)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + alpha) + 2.0)) / Float64(2.0 + Float64(1.0 + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+93], N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000009e93

   1. Initial program 99.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 94.5%

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

   if 2.00000000000000009e93 < beta

   1. Initial program 84.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 92.5

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

   7. Simplified 92.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t\_0 \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+92)
     (/
      (+ alpha (+ beta (fma alpha beta 1.0)))
      (* t_0 (* (+ alpha (+ beta 3.0)) t_0)))
     (/
      (/ (+ alpha 1.0) (+ (+ beta alpha) 2.0))
      (+ 2.0 (+ 1.0 (+ beta alpha)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+92) {
		tmp = (alpha + (beta + fma(alpha, beta, 1.0))) / (t_0 * ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+92)
		tmp = Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / Float64(t_0 * Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + alpha) + 2.0)) / Float64(2.0 + Float64(1.0 + 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+92], N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e92

   1. Initial program 99.3%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.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. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 94.4%

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

   if 2.0000000000000001e92 < beta

   1. Initial program 84.6%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 84.6

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 92.5

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

   7. Simplified 92.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 2 + \left(1 + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{t\_0}\\ \end{array} \end{array} \]

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 (+ 1.0 (+ beta alpha)))))
   (if (<= beta 2500000000.0)
     (/ (/ (+ beta 1.0) (* (+ beta 2.0) (+ beta 2.0))) t_0)
     (/ (/ (+ alpha 1.0) (+ (+ beta alpha) 2.0)) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 2.0 + (1.0 + (beta + alpha));
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / 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 + (1.0d0 + (beta + alpha))
    if (beta <= 2500000000.0d0) then
        tmp = ((beta + 1.0d0) / ((beta + 2.0d0) * (beta + 2.0d0))) / t_0
    else
        tmp = ((alpha + 1.0d0) / ((beta + alpha) + 2.0d0)) / 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 + (1.0 + (beta + alpha));
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 2.0 + (1.0 + (beta + alpha))
	tmp = 0
	if beta <= 2500000000.0:
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_0
	else:
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(1.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 2500000000.0)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + alpha) + 2.0)) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (1.0 + (beta + alpha));
	tmp = 0.0;
	if (beta <= 2500000000.0)
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_0;
	else
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / 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[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2500000000.0], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e9

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 67.8

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

   7. Simplified 67.8%

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

   if 2.5e9 < beta

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   5. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 85.1

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

   7. Simplified 85.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2500000000.0)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/
    (/ (+ alpha 1.0) (+ (+ beta alpha) 2.0))
    (+ 2.0 (+ 1.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2500000000.0d0) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / ((beta + alpha) + 2.0d0)) / (2.0d0 + (1.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2500000000.0:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2500000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + alpha) + 2.0)) / Float64(2.0 + Float64(1.0 + Float64(beta + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2500000000.0)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / ((beta + alpha) + 2.0)) / (2.0 + (1.0 + (beta + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2500000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e9

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.6

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

   5. Simplified 66.6%

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

   if 2.5e9 < beta

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   5. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 85.1

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

   7. Simplified 85.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + \alpha\right) + 2}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2500000000.0)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ 2.0 (+ 1.0 (+ beta alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2500000000.0d0) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (2.0d0 + (1.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (2.0 + (1.0 + (beta + alpha)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2500000000.0:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (2.0 + (1.0 + (beta + alpha)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2500000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(2.0 + Float64(1.0 + Float64(beta + alpha))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2500000000.0)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (2.0 + (1.0 + (beta + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2500000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 + N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e9

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.6

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

   5. Simplified 66.6%

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

   if 2.5e9 < beta

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   5. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 84.5

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

   7. Simplified 84.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{2 + \left(1 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta}}{-2 - \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2500000000.0)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (- -1.0 alpha) beta) (- -2.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((-1.0 - alpha) / beta) / (-2.0 - (beta + alpha));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2500000000.0d0) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = (((-1.0d0) - alpha) / beta) / ((-2.0d0) - (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((-1.0 - alpha) / beta) / (-2.0 - (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2500000000.0:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((-1.0 - alpha) / beta) / (-2.0 - (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2500000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(-1.0 - alpha) / beta) / Float64(-2.0 - Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2500000000.0)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((-1.0 - alpha) / beta) / (-2.0 - (beta + alpha));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2500000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e9

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.6

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

   5. Simplified 66.6%

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

   if 2.5e9 < beta

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   6. Applied egg-rr 73.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

   8. Applied egg-rr 84.1%

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

   9. Taylor expanded in beta around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. --lowering--.f64 84.5

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

   11. Simplified 84.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta}}{-2 - \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2500000000.0)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2500000000.0d0) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2500000000.0) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2500000000.0:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2500000000.0)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2500000000.0)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2500000000.0], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5e9

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 66.6

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

   5. Simplified 66.6%

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

   if 2.5e9 < beta

   1. Initial program 88.4%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 81.2

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

   5. Simplified 81.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 84.3

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2500000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.55:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.55)
   (/ (+ alpha 1.0) (* (* (+ alpha 2.0) (+ alpha 2.0)) (+ alpha 3.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.55) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.55d0) then
        tmp = (alpha + 1.0d0) / (((alpha + 2.0d0) * (alpha + 2.0d0)) * (alpha + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.55) {
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.55:
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.55)
		tmp = Float64(Float64(alpha + 1.0) / Float64(Float64(Float64(alpha + 2.0) * Float64(alpha + 2.0)) * Float64(alpha + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.55)
		tmp = (alpha + 1.0) / (((alpha + 2.0) * (alpha + 2.0)) * (alpha + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.55], N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5499999999999998

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 92.6

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

   5. Simplified 92.6%

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

   if 3.5499999999999998 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 82.4

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

   7. Applied egg-rr 82.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.55:\\ \;\;\;\;\frac{\alpha + 1}{\left(\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha \cdot \alpha, -0.0625, 0.25\right)}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ (fma (* alpha alpha) -0.0625 0.25) (- (+ beta alpha) -3.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = fma((alpha * alpha), -0.0625, 0.25) / ((beta + alpha) - -3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(fma(Float64(alpha * alpha), -0.0625, 0.25) / Float64(Float64(beta + alpha) - -3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(N[(N[(alpha * alpha), $MachinePrecision] * -0.0625 + 0.25), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} + {\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right) + \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\frac{1}{16} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\alpha \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \alpha \cdot \frac{1}{16} + \color{blue}{\frac{-1}{16}}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. accelerator-lowering-fma.f64 65.6

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.0625, -0.0625\right)}, 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right)\right)\right)} + \left(2 + 1\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right)\right)\right) + \color{blue}{3}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)}} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right) + -3\right)\right)}} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{0 - \left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right) + -3\right)}} \]

      8. associate--r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right)\right) - -3}} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\alpha + \beta\right)\right)\right)\right)\right)} - -3} \]

      10. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\alpha + \beta\right)} - -3} \]

      11. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\alpha + \beta\right) - -3}} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, \frac{1}{16}, \frac{-1}{16}\right), \frac{1}{4}\right)}{\color{blue}{\left(\beta + \alpha\right)} - -3} \]

      13. +-lowering-+.f64 65.6

          \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}{\color{blue}{\left(\beta + \alpha\right)} - -3} \]

   10. Applied egg-rr 65.6%

       \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}{\color{blue}{\left(\beta + \alpha\right) - -3}} \]

   11. Taylor expanded in alpha around 0

       \[\leadsto \frac{\color{blue}{\frac{1}{4} + \frac{-1}{16} \cdot {\alpha}^{2}}}{\left(\beta + \alpha\right) - -3} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot {\alpha}^{2} + \frac{1}{4}}}{\left(\beta + \alpha\right) - -3} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \frac{-1}{16}} + \frac{1}{4}}{\left(\beta + \alpha\right) - -3} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{-1}{16}, \frac{1}{4}\right)}}{\left(\beta + \alpha\right) - -3} \]

       4. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{-1}{16}, \frac{1}{4}\right)}{\left(\beta + \alpha\right) - -3} \]

       5. *-lowering-*.f64 65.1

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, -0.0625, 0.25\right)}{\left(\beta + \alpha\right) - -3} \]

   13. Simplified 65.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, -0.0625, 0.25\right)}}{\left(\beta + \alpha\right) - -3} \]

   if 5.20000000000000018 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 82.4

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

   7. Applied egg-rr 82.4%

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

Alternative 16: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.25}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6)
   (/ 0.25 (- (+ beta alpha) -3.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.6d0) then
        tmp = 0.25d0 / ((beta + alpha) - (-3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = 0.25 / ((beta + alpha) - -3.0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(0.25 / Float64(Float64(beta + alpha) - -3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = 0.25 / ((beta + alpha) - -3.0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(0.25 / N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 66.2%

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

      1. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 66.2

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

   10. Applied egg-rr 66.2%

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

   if 6.5999999999999996 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 82.4

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

   7. Applied egg-rr 82.4%

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

Alternative 17: 94.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.25 (- (+ beta alpha) -3.0))
   (* (+ alpha 1.0) (/ 1.0 (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / ((beta + alpha) - (-3.0d0))
    else
        tmp = (alpha + 1.0d0) * (1.0d0 / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / ((beta + alpha) - -3.0)
	else:
		tmp = (alpha + 1.0) * (1.0 / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(Float64(beta + alpha) - -3.0));
	else
		tmp = Float64(Float64(alpha + 1.0) * Float64(1.0 / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / ((beta + alpha) - -3.0);
	else
		tmp = (alpha + 1.0) * (1.0 / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 66.2%

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

      1. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 66.2

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

   10. Applied egg-rr 66.2%

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

   if 6 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 79.4

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

   7. Applied egg-rr 79.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ 0.25 (- (+ beta alpha) -3.0))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 0.25d0 / ((beta + alpha) - (-3.0d0))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 0.25 / ((beta + alpha) - -3.0);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 0.25 / ((beta + alpha) - -3.0)
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(0.25 / Float64(Float64(beta + alpha) - -3.0));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 0.25 / ((beta + alpha) - -3.0);
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(0.25 / N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 66.2%

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

      1. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 66.2

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

   10. Applied egg-rr 66.2%

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

   if 6.5 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{0.25}{\left(\beta + \alpha\right) - -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 66.2%

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

      1. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 66.2

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

   10. Applied egg-rr 66.2%

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

   if 6.20000000000000018 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

   6. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

Alternative 20: 91.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6) (/ 0.25 (+ alpha 3.0)) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.6], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 66.2%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f64 65.6

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

   11. Simplified 65.6%

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

   if 3.60000000000000009 < beta

   1. Initial program 88.8%

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

   3. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

   6. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

Alternative 21: 55.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.009259259259259259, -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot 0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.45e+41)
   (fma
    beta
    (fma beta 0.009259259259259259 -0.027777777777777776)
    0.08333333333333333)
   (* (* alpha alpha) 0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.45e+41) {
		tmp = fma(beta, fma(beta, 0.009259259259259259, -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = (alpha * alpha) * 0.0625;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.45e+41)
		tmp = fma(beta, fma(beta, 0.009259259259259259, -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha * alpha) * 0.0625);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.45e+41], N[(beta * N[(beta * 0.009259259259259259 + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(N[(alpha * alpha), $MachinePrecision] * 0.0625), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.44999999999999994e41

   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

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in alpha around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{1}{108} \cdot \beta - \frac{1}{36}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\beta \cdot \left(\frac{1}{108} \cdot \beta - \frac{1}{36}\right) + \frac{1}{12}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{108} \cdot \beta - \frac{1}{36}, \frac{1}{12}\right)} \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{1}{108} \cdot \beta + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{1}{108}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\beta, \beta \cdot \frac{1}{108} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

       6. accelerator-lowering-fma.f64 61.0

          \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, 0.009259259259259259, -0.027777777777777776\right)}, 0.08333333333333333\right) \]

   11. Simplified 61.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.009259259259259259, -0.027777777777777776\right), 0.08333333333333333\right)} \]

   if 1.44999999999999994e41 < beta

   1. Initial program 87.2%

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

   3. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-lowering-+.f64 16.2

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

   5. Simplified 16.2%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{4} + {\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{\alpha}^{2} \cdot \left(\frac{1}{16} \cdot \alpha - \frac{1}{16}\right) + \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\alpha}^{2}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha}, \frac{1}{16} \cdot \alpha - \frac{1}{16}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\frac{1}{16} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\alpha \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \alpha \cdot \frac{1}{16} + \color{blue}{\frac{-1}{16}}, \frac{1}{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. accelerator-lowering-fma.f64 5.6

         \[\leadsto \frac{\mathsf{fma}\left(\alpha \cdot \alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.0625, -0.0625\right)}, 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. Simplified 5.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha, \mathsf{fma}\left(\alpha, 0.0625, -0.0625\right), 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot {\alpha}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{\alpha}^{2} \cdot \frac{1}{16}} \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \frac{1}{16} \]

       4. *-lowering-*.f64 23.9

          \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot 0.0625 \]

   11. Simplified 23.9%

       \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot 0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 44.7% accurate, 84.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in beta around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-lowering-+.f64 69.7

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

5. Simplified 69.7%

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

6. Taylor expanded in alpha around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. +-lowering-+.f64 44.6

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

8. Simplified 44.6%

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

9. Taylor expanded in beta around 0

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

11. Simplified 43.2%

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

Reproduce

herbie shell --seed 2024199 
(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)))