Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.7%

Time: 14.0s

Alternatives: 27

Speedup: 2.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}\right)}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ 1.0 t_0)))
   (if (<= beta 2e+135)
     (/ (/ (/ (+ (+ (+ beta alpha) (* beta alpha)) 1.0) t_0) t_0) t_1)
     (/
      (/
       (+
        (+ (/ 1.0 beta) (+ alpha (/ alpha beta)))
        (+ 1.0 (* (- -1.0 alpha) (/ (+ alpha 2.0) beta))))
       t_0)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 2e+135) {
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	} else {
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * ((alpha + 2.0) / beta)))) / t_0) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = 1.0d0 + t_0
    if (beta <= 2d+135) then
        tmp = (((((beta + alpha) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / t_1
    else
        tmp = ((((1.0d0 / beta) + (alpha + (alpha / beta))) + (1.0d0 + (((-1.0d0) - alpha) * ((alpha + 2.0d0) / beta)))) / t_0) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 2e+135) {
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	} else {
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * ((alpha + 2.0) / beta)))) / t_0) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = 1.0 + t_0
	tmp = 0
	if beta <= 2e+135:
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1
	else:
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * ((alpha + 2.0) / beta)))) / t_0) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (beta <= 2e+135)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + alpha) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha + Float64(alpha / beta))) + Float64(1.0 + Float64(Float64(-1.0 - alpha) * Float64(Float64(alpha + 2.0) / beta)))) / t_0) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = 1.0 + t_0;
	tmp = 0.0;
	if (beta <= 2e+135)
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	else
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * ((alpha + 2.0) / beta)))) / t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2e+135], N[(N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $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[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999992e135

   1. Initial program 99.4%

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

   if 1.99999999999999992e135 < beta

   1. Initial program 66.5%

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

   3. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      12. associate-/l* N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

   5. Simplified 86.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}\right)}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-177}:\\ \;\;\;\;0.024691358024691357 \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1
         (/
          (/ (/ (+ (+ (+ beta alpha) (* beta alpha)) 1.0) t_0) t_0)
          (+ 1.0 t_0))))
   (if (<= t_1 2e-177)
     (* 0.024691358024691357 (* alpha (* alpha alpha)))
     (if (<= t_1 0.1) (/ 0.25 (+ alpha 3.0)) (/ 1.0 (* alpha alpha))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	double tmp;
	if (t_1 <= 2e-177) {
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	} else if (t_1 <= 0.1) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (alpha * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = (((((beta + alpha) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (1.0d0 + t_0)
    if (t_1 <= 2d-177) then
        tmp = 0.024691358024691357d0 * (alpha * (alpha * alpha))
    else if (t_1 <= 0.1d0) then
        tmp = 0.25d0 / (alpha + 3.0d0)
    else
        tmp = 1.0d0 / (alpha * alpha)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	double tmp;
	if (t_1 <= 2e-177) {
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	} else if (t_1 <= 0.1) {
		tmp = 0.25 / (alpha + 3.0);
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0)
	tmp = 0
	if t_1 <= 2e-177:
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha))
	elif t_1 <= 0.1:
		tmp = 0.25 / (alpha + 3.0)
	else:
		tmp = 1.0 / (alpha * alpha)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(beta + alpha) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(1.0 + t_0))
	tmp = 0.0
	if (t_1 <= 2e-177)
		tmp = Float64(0.024691358024691357 * Float64(alpha * Float64(alpha * alpha)));
	elseif (t_1 <= 0.1)
		tmp = Float64(0.25 / Float64(alpha + 3.0));
	else
		tmp = Float64(1.0 / Float64(alpha * alpha));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	tmp = 0.0;
	if (t_1 <= 2e-177)
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	elseif (t_1 <= 0.1)
		tmp = 0.25 / (alpha + 3.0);
	else
		tmp = 1.0 / (alpha * alpha);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-177], N[(0.024691358024691357 * N[(alpha * N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < 1.9999999999999999e-177

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 46.3

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

   5. Simplified 46.3%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 2.3

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

   8. Simplified 2.3%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{2}{81} \cdot {\alpha}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{3} \cdot \frac{2}{81}} \]

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

          \[\leadsto \color{blue}{{\alpha}^{3} \cdot \frac{2}{81}} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)} \cdot \frac{2}{81} \]

       4. unpow2 N/A

          \[\leadsto \left(\alpha \cdot \color{blue}{{\alpha}^{2}}\right) \cdot \frac{2}{81} \]

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

          \[\leadsto \color{blue}{\left(\alpha \cdot {\alpha}^{2}\right)} \cdot \frac{2}{81} \]

       6. unpow2 N/A

          \[\leadsto \left(\alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot \frac{2}{81} \]

       7. *-lowering-*.f64 24.3

          \[\leadsto \left(\alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot 0.024691358024691357 \]

   11. Simplified 24.3%

       \[\leadsto \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right) \cdot 0.024691358024691357} \]

   if 1.9999999999999999e-177 < (/.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))) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 88.0

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

   5. Simplified 88.0%

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

   6. Taylor expanded in beta around 0

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

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

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

      2. +-lowering-+.f64 78.5

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

   8. Simplified 78.5%

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

   if 0.10000000000000001 < (/.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 1.6%

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

   3. Taylor expanded in beta around 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 49.2

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

   5. Simplified 49.2%

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

   6. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 44.0

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

   8. Simplified 44.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 2 \cdot 10^{-177}:\\ \;\;\;\;0.024691358024691357 \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 0.1:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ 1.0 t_0)))
   (if (<= beta 2e+135)
     (/ (/ (/ (+ (+ (+ beta alpha) (* beta alpha)) 1.0) t_0) t_0) t_1)
     (/
      (/
       (+
        (+ (+ alpha 1.0) (+ (/ 1.0 beta) (/ alpha beta)))
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 2e+135) {
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	} else {
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / t_1;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (beta <= 2e+135)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + alpha) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + 1.0) + Float64(Float64(1.0 / beta) + Float64(alpha / beta))) + Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / t_1);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2e+135], N[(N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] + N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999992e135

   1. Initial program 99.4%

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

   if 1.99999999999999992e135 < beta

   1. Initial program 66.5%

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

   3. Taylor expanded in beta around inf

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

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

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

   5. Simplified 86.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{1 + t\_0} \leq 2 \cdot 10^{-177}:\\ \;\;\;\;0.024691358024691357 \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \end{array} \end{array} \]

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 (<=
        (/
         (/ (/ (+ (+ (+ beta alpha) (* beta alpha)) 1.0) t_0) t_0)
         (+ 1.0 t_0))
        2e-177)
     (* 0.024691358024691357 (* alpha (* alpha alpha)))
     (/ 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 (((((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0)) <= 2e-177) {
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	} 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 (((((((beta + alpha) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (1.0d0 + t_0)) <= 2d-177) then
        tmp = 0.024691358024691357d0 * (alpha * (alpha * alpha))
    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 (((((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0)) <= 2e-177) {
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	} 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 ((((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0)) <= 2e-177:
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha))
	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(Float64(Float64(beta + alpha) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(1.0 + t_0)) <= 2e-177)
		tmp = Float64(0.024691358024691357 * Float64(alpha * Float64(alpha * alpha)));
	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 (((((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / (1.0 + t_0)) <= 2e-177)
		tmp = 0.024691358024691357 * (alpha * (alpha * alpha));
	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[(N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 2e-177], N[(0.024691358024691357 * N[(alpha * N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $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))) < 1.9999999999999999e-177

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 46.3

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

   5. Simplified 46.3%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 2.3

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

   8. Simplified 2.3%

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

   9. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{2}{81} \cdot {\alpha}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{\alpha}^{3} \cdot \frac{2}{81}} \]

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

          \[\leadsto \color{blue}{{\alpha}^{3} \cdot \frac{2}{81}} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)} \cdot \frac{2}{81} \]

       4. unpow2 N/A

          \[\leadsto \left(\alpha \cdot \color{blue}{{\alpha}^{2}}\right) \cdot \frac{2}{81} \]

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

          \[\leadsto \color{blue}{\left(\alpha \cdot {\alpha}^{2}\right)} \cdot \frac{2}{81} \]

       6. unpow2 N/A

          \[\leadsto \left(\alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot \frac{2}{81} \]

       7. *-lowering-*.f64 24.3

          \[\leadsto \left(\alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot 0.024691358024691357 \]

   11. Simplified 24.3%

       \[\leadsto \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right) \cdot 0.024691358024691357} \]

   if 1.9999999999999999e-177 < (/.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 89.3%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in beta around 0

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

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

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

      2. +-lowering-+.f64 70.8

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

   8. Simplified 70.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)} \leq 2 \cdot 10^{-177}:\\ \;\;\;\;0.024691358024691357 \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \left(-1 + \frac{\alpha + 2}{\beta}\right)}{\beta}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (+ 1.0 t_0)))
   (if (<= beta 2e+135)
     (/ (/ (/ (+ (+ (+ beta alpha) (* beta alpha)) 1.0) t_0) t_0) t_1)
     (/ (/ (* (- -1.0 alpha) (+ -1.0 (/ (+ alpha 2.0) beta))) beta) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 2e+135) {
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	} else {
		tmp = (((-1.0 - alpha) * (-1.0 + ((alpha + 2.0) / beta))) / beta) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = 1.0d0 + t_0
    if (beta <= 2d+135) then
        tmp = (((((beta + alpha) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / t_1
    else
        tmp = ((((-1.0d0) - alpha) * ((-1.0d0) + ((alpha + 2.0d0) / beta))) / beta) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 2e+135) {
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	} else {
		tmp = (((-1.0 - alpha) * (-1.0 + ((alpha + 2.0) / beta))) / beta) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = 1.0 + t_0
	tmp = 0
	if beta <= 2e+135:
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1
	else:
		tmp = (((-1.0 - alpha) * (-1.0 + ((alpha + 2.0) / beta))) / beta) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (beta <= 2e+135)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + alpha) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(-1.0 + Float64(Float64(alpha + 2.0) / beta))) / beta) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = 1.0 + t_0;
	tmp = 0.0;
	if (beta <= 2e+135)
		tmp = (((((beta + alpha) + (beta * alpha)) + 1.0) / t_0) / t_0) / t_1;
	else
		tmp = (((-1.0 - alpha) * (-1.0 + ((alpha + 2.0) / beta))) / beta) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2e+135], N[(N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(-1.0 + N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999992e135

   1. Initial program 99.4%

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

   if 1.99999999999999992e135 < beta

   1. Initial program 66.5%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.6

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

   5. Simplified 86.6%

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

   6. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

      17. neg-lowering-neg.f64 85.9

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

   8. Simplified 85.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \left(-1 + \frac{\alpha + 2}{\beta}\right)}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t\_0}}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \left(-1 + \frac{\alpha + 2}{\beta}\right)}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 4e+135)
     (/
      (/ (+ alpha (+ beta (fma alpha beta 1.0))) t_0)
      (* t_0 (+ alpha (+ beta 3.0))))
     (/
      (/ (* (- -1.0 alpha) (+ -1.0 (/ (+ alpha 2.0) beta))) beta)
      (+ 1.0 (+ (+ beta alpha) 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 4e+135) {
		tmp = ((alpha + (beta + fma(alpha, beta, 1.0))) / t_0) / (t_0 * (alpha + (beta + 3.0)));
	} else {
		tmp = (((-1.0 - alpha) * (-1.0 + ((alpha + 2.0) / beta))) / beta) / (1.0 + ((beta + alpha) + 2.0));
	}
	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 <= 4e+135)
		tmp = Float64(Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 - alpha) * Float64(-1.0 + Float64(Float64(alpha + 2.0) / beta))) / beta) / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4e+135], N[(N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] * N[(-1.0 + N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.99999999999999985e135

   1. Initial program 99.4%

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

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

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

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

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.3%

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

   if 3.99999999999999985e135 < beta

   1. Initial program 66.5%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.6

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

   5. Simplified 86.6%

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

   6. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

      17. neg-lowering-neg.f64 85.9

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

   8. Simplified 85.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 - \alpha\right) \cdot \left(-1 + \frac{\alpha + 2}{\beta}\right)}{\beta}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{t\_0}}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.4e+135)
     (/
      (/ (+ alpha (+ beta (fma alpha beta 1.0))) t_0)
      (* t_0 (+ alpha (+ beta 3.0))))
     (/ (/ (+ alpha 1.0) (+ beta (+ alpha 3.0))) (+ (+ beta alpha) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.4e+135) {
		tmp = ((alpha + (beta + fma(alpha, beta, 1.0))) / t_0) / (t_0 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / ((beta + alpha) + 2.0);
	}
	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 <= 2.4e+135)
		tmp = Float64(Float64(Float64(alpha + Float64(beta + fma(alpha, beta, 1.0))) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) / Float64(Float64(beta + alpha) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.4e+135], N[(N[(N[(alpha + N[(beta + N[(alpha * beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.39999999999999997e135

   1. Initial program 99.4%

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

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

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

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

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.3%

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

   if 2.39999999999999997e135 < beta

   1. Initial program 66.5%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.6

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

   5. Simplified 86.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

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

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 86.6

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

   7. Applied egg-rr 86.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1}{t\_0} \cdot \left(\beta + 1\right)}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 3.0))))
   (if (<= beta 3.3e+16)
     (/ (* (/ 1.0 t_0) (+ beta 1.0)) (* (+ beta 2.0) (+ beta 2.0)))
     (/ (/ (+ alpha 1.0) t_0) (+ (+ beta alpha) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 3.0);
	double tmp;
	if (beta <= 3.3e+16) {
		tmp = ((1.0 / t_0) * (beta + 1.0)) / ((beta + 2.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / ((beta + alpha) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 3.0d0)
    if (beta <= 3.3d+16) then
        tmp = ((1.0d0 / t_0) * (beta + 1.0d0)) / ((beta + 2.0d0) * (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / t_0) / ((beta + alpha) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 3.0);
	double tmp;
	if (beta <= 3.3e+16) {
		tmp = ((1.0 / t_0) * (beta + 1.0)) / ((beta + 2.0) * (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / ((beta + alpha) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = beta + (alpha + 3.0)
	tmp = 0
	if beta <= 3.3e+16:
		tmp = ((1.0 / t_0) * (beta + 1.0)) / ((beta + 2.0) * (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / t_0) / ((beta + alpha) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 3.0))
	tmp = 0.0
	if (beta <= 3.3e+16)
		tmp = Float64(Float64(Float64(1.0 / t_0) * Float64(beta + 1.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(Float64(beta + alpha) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 3.0);
	tmp = 0.0;
	if (beta <= 3.3e+16)
		tmp = ((1.0 / t_0) * (beta + 1.0)) / ((beta + 2.0) * (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / t_0) / ((beta + alpha) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.3e+16], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

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

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

      14. +-commutative N/A

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

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

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

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

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

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

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

      18. +-lowering-+.f64 66.7

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

   7. Applied egg-rr 66.7%

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

   if 3.3e16 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

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

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 86.4

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1}{\beta + \left(\alpha + 3\right)} \cdot \left(\beta + 1\right)}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 6.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 6.9e+15)
     (/ (/ (+ beta 1.0) (* (+ beta 2.0) (+ beta 2.0))) (+ 1.0 t_0))
     (/ (/ (+ alpha 1.0) (+ beta (+ alpha 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 6.9e+15) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / 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 = (beta + alpha) + 2.0d0
    if (beta <= 6.9d+15) then
        tmp = ((beta + 1.0d0) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + t_0)
    else
        tmp = ((alpha + 1.0d0) / (beta + (alpha + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 6.9e+15) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 6.9e+15:
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0)
	else:
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 6.9e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 6.9e+15)
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + t_0);
	else
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 6.9e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.9e15

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

   if 6.9e15 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

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

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 86.4

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.9e+15)
   (/ (+ beta 1.0) (* (fma beta (+ beta 4.0) 4.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) (+ beta (+ alpha 3.0))) (+ (+ beta alpha) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.9e+15) {
		tmp = (beta + 1.0) / (fma(beta, (beta + 4.0), 4.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / ((beta + alpha) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.9e+15)
		tmp = Float64(Float64(beta + 1.0) / Float64(fma(beta, Float64(beta + 4.0), 4.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) / Float64(Float64(beta + alpha) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.9e+15], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.9e15

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   7. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-lowering-+.f64 65.1

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

   8. Simplified 65.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 65.1

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

   11. Simplified 65.1%

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

   if 7.9e15 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

      1. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

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

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

      18. +-commutative N/A

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

      19. +-lowering-+.f64 86.4

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\left(\beta + \alpha\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.1)
   (/ 0.25 (+ 1.0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.35e+154)
     (/ (+ alpha 1.0) (* beta (+ beta (+ alpha 3.0))))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (beta * (beta + (alpha + 3.0)));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.1d0) then
        tmp = 0.25d0 / (1.0d0 + ((beta + alpha) + 2.0d0))
    else if (beta <= 1.35d+154) then
        tmp = (alpha + 1.0d0) / (beta * (beta + (alpha + 3.0d0)))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (beta * (beta + (alpha + 3.0)));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.1:
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0))
	elif beta <= 1.35e+154:
		tmp = (alpha + 1.0) / (beta * (beta + (alpha + 3.0)))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.1)
		tmp = Float64(0.25 / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * Float64(beta + Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.1)
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	elseif (beta <= 1.35e+154)
		tmp = (alpha + 1.0) / (beta * (beta + (alpha + 3.0)));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.1], N[(0.25 / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.0999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 66.9%

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

   if 4.0999999999999996 < beta < 1.35000000000000003e154

   1. Initial program 96.6%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. +-lowering-+.f64 85.6

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

   5. Simplified 85.6%

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 88.7

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

   7. Applied egg-rr 88.7%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 62.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 79.9

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

   8. Simplified 79.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 82.5

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.56e+16)
   (/ (+ beta 1.0) (* (fma beta (+ beta 4.0) 4.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) (+ beta (+ alpha 3.0))) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.56e+16) {
		tmp = (beta + 1.0) / (fma(beta, (beta + 4.0), 4.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (beta + (alpha + 3.0))) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.56e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(fma(beta, Float64(beta + 4.0), 4.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(beta + Float64(alpha + 3.0))) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.56e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   7. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-lowering-+.f64 65.1

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

   8. Simplified 65.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 65.1

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

   11. Simplified 65.1%

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

   if 1.56e16 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. +-lowering-+.f64 86.0

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

   5. Simplified 86.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 86.0

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

   7. Applied egg-rr 86.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta + \left(\alpha + 3\right)}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.1e+16)
   (/ (+ beta 1.0) (* (fma beta (+ beta 4.0) 4.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.1e+16) {
		tmp = (beta + 1.0) / (fma(beta, (beta + 4.0), 4.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.1e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(fma(beta, Float64(beta + 4.0), 4.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.1e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.1e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   7. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-lowering-+.f64 65.1

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

   8. Simplified 65.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. +-lowering-+.f64 65.1

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

   11. Simplified 65.1%

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

   if 6.1e16 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 83.1

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

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. +-lowering-+.f64 85.9

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.66 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.66e+16)
   (/ (+ beta 1.0) (fma beta (fma beta (+ beta 7.0) 16.0) 12.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.66e+16) {
		tmp = (beta + 1.0) / fma(beta, fma(beta, (beta + 7.0), 16.0), 12.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.66e+16)
		tmp = Float64(Float64(beta + 1.0) / fma(beta, fma(beta, Float64(beta + 7.0), 16.0), 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.66e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta * N[(beta + 7.0), $MachinePrecision] + 16.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.66e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   7. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-lowering-+.f64 65.1

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

   8. Simplified 65.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. +-lowering-+.f64 65.1

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

   11. Simplified 65.1%

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

   if 1.66e16 < beta

   1. Initial program 76.3%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 83.1

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

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. +-lowering-+.f64 85.9

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.66 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\alpha + 1}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5)
   (/ (+ alpha 1.0) (fma alpha (fma alpha (+ alpha 7.0) 16.0) 12.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = (alpha + 1.0) / fma(alpha, fma(alpha, (alpha + 7.0), 16.0), 12.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(Float64(alpha + 1.0) / fma(alpha, fma(alpha, Float64(alpha + 7.0), 16.0), 12.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 9.5], N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha * N[(alpha * N[(alpha + 7.0), $MachinePrecision] + 16.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 92.9

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

   8. Simplified 92.9%

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

   if 9.5 < beta

   1. Initial program 76.9%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 82.2

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

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. +-lowering-+.f64 84.9

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\alpha + 1}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha + 7, 16\right), 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ 1.0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 1.35e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.2d0) then
        tmp = 0.25d0 / (1.0d0 + ((beta + alpha) + 2.0d0))
    else if (beta <= 1.35d+154) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	} else if (beta <= 1.35e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0))
	elif beta <= 1.35e+154:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(1.0 + Float64(Float64(beta + alpha) + 2.0)));
	elseif (beta <= 1.35e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (1.0 + ((beta + alpha) + 2.0));
	elseif (beta <= 1.35e+154)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.35e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.20000000000000018

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 66.9%

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

   if 6.20000000000000018 < beta < 1.35000000000000003e154

   1. Initial program 96.6%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 85.4

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

   5. Simplified 85.4%

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

   if 1.35000000000000003e154 < beta

   1. Initial program 62.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   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 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 79.9

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

   8. Simplified 79.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 82.5

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 96.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (if (<= beta 1.7e+154)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else if (beta <= 1.7e+154) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	elseif (beta <= 1.7e+154)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], If[LessEqual[beta, 1.7e+154], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 65.3

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

   8. Simplified 65.3%

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

   if 3.39999999999999991 < beta < 1.69999999999999987e154

   1. Initial program 93.7%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 85.9

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

   5. Simplified 85.9%

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

   if 1.69999999999999987e154 < beta

   1. Initial program 64.2%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 79.5

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

   5. Simplified 79.5%

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

   6. Taylor expanded in alpha around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 79.5

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

   8. Simplified 79.5%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 84.3

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

   10. Applied egg-rr 84.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.0)
   (/ 0.25 (+ 1.0 (+ (+ beta alpha) 2.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.0) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 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 <= 7.0d0) then
        tmp = 0.25d0 / (1.0d0 + ((beta + alpha) + 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 <= 7.0) {
		tmp = 0.25 / (1.0 + ((beta + alpha) + 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 <= 7.0:
		tmp = 0.25 / (1.0 + ((beta + alpha) + 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 <= 7.0)
		tmp = Float64(0.25 / Float64(1.0 + Float64(Float64(beta + alpha) + 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 <= 7.0)
		tmp = 0.25 / (1.0 + ((beta + alpha) + 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, 7.0], N[(0.25 / N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 66.9%

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

   if 7 < beta

   1. Initial program 76.9%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 82.2

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

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   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. +-lowering-+.f64 84.9

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 94.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 65.3

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

   8. Simplified 65.3%

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

   if 3.39999999999999991 < beta

   1. Initial program 76.9%

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

   3. Taylor expanded in beta around inf

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

      1. /-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 82.2

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

   5. Simplified 82.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 91.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.15)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.15)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.15], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.14999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, \frac{2}{81} \cdot \alpha - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. accelerator-lowering-fma.f64 65.3

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right)}, -0.027777777777777776\right), 0.08333333333333333\right) \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)} \]

   if 2.14999999999999991 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-lowering-+.f64 85.1

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Simplified 85.1%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \left(3 + \beta\right)}} \]

      3. +-lowering-+.f64 75.1

         \[\leadsto \frac{1}{\beta \cdot \color{blue}{\left(3 + \beta\right)}} \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(3 + \beta\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 91.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (fma
    alpha
    (fma
     alpha
     (fma alpha 0.024691358024691357 -0.011574074074074073)
     -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = fma(alpha, fma(alpha, fma(alpha, 0.024691358024691357, -0.011574074074074073), -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(alpha * N[(alpha * N[(alpha * 0.024691358024691357 + -0.011574074074074073), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 92.9

         \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, \frac{2}{81} \cdot \alpha - \frac{5}{432}, \frac{-1}{36}\right)}, \frac{1}{12}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\frac{2}{81} \cdot \alpha + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{2}{81}} + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right), \frac{-1}{36}\right), \frac{1}{12}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \alpha \cdot \frac{2}{81} + \color{blue}{\frac{-5}{432}}, \frac{-1}{36}\right), \frac{1}{12}\right) \]

      9. accelerator-lowering-fma.f64 65.3

         \[\leadsto \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right)}, -0.027777777777777776\right), 0.08333333333333333\right) \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.024691358024691357, -0.011574074074074073\right), -0.027777777777777776\right), 0.08333333333333333\right)} \]

   if 3.39999999999999991 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{{\left(2 + \beta\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. +-lowering-+.f64 76.5

         \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Simplified 76.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   7. Step-by-step derivation

      1. /-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 75.1

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 91.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.4)
   (fma
    alpha
    (fma alpha -0.011574074074074073 -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.4) {
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.4)
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.4], N[(alpha * N[(alpha * -0.011574074074074073 + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999991

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 92.9

         \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{-5}{432} \cdot \alpha - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{-5}{432} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{-5}{432}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{-5}{432} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      6. accelerator-lowering-fma.f64 64.8

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right)}, 0.08333333333333333\right) \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)} \]

   if 3.39999999999999991 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{{\left(2 + \beta\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      8. +-lowering-+.f64 76.5

         \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Simplified 76.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   7. Step-by-step derivation

      1. /-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 75.1

         \[\leadsto \frac{1}{\color{blue}{\beta \cdot \beta}} \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 46.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.0)
   (fma
    alpha
    (fma alpha -0.011574074074074073 -0.027777777777777776)
    0.08333333333333333)
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.0) {
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.0)
		tmp = fma(alpha, fma(alpha, -0.011574074074074073, -0.027777777777777776), 0.08333333333333333);
	else
		tmp = Float64(1.0 / 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, 9.0], N[(alpha * N[(alpha * -0.011574074074074073 + -0.027777777777777776), $MachinePrecision] + 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 92.9

         \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right) + \frac{1}{12}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{-5}{432} \cdot \alpha - \frac{1}{36}, \frac{1}{12}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{-5}{432} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{12}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{-5}{432}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{12}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{-5}{432} + \color{blue}{\frac{-1}{36}}, \frac{1}{12}\right) \]

      6. accelerator-lowering-fma.f64 64.8

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right)}, 0.08333333333333333\right) \]

   8. Simplified 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.011574074074074073, -0.027777777777777776\right), 0.08333333333333333\right)} \]

   if 9 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-lowering-+.f64 85.1

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Simplified 85.1%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 7.1

         \[\leadsto \color{blue}{\frac{1}{\beta}} \]

   8. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 46.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.0)
   (fma alpha -0.027777777777777776 0.08333333333333333)
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = fma(alpha, -0.027777777777777776, 0.08333333333333333);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.0)
		tmp = fma(alpha, -0.027777777777777776, 0.08333333333333333);
	else
		tmp = Float64(1.0 / 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, 10.0], N[(alpha * -0.027777777777777776 + 0.08333333333333333), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10

   1. Initial program 99.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 92.9

         \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{36} \cdot \alpha + \frac{1}{12}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \frac{-1}{36}} + \frac{1}{12} \]

      3. accelerator-lowering-fma.f64 64.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)} \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)} \]

   if 10 < beta

   1. Initial program 76.9%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

      2. +-lowering-+.f64 85.1

         \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. Simplified 85.1%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 7.1

         \[\leadsto \color{blue}{\frac{1}{\beta}} \]

   8. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 45.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.25}{\alpha + 3} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ alpha 3.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (alpha + 3.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.25 / (alpha + 3.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.25 / (alpha + 3.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.25 / Float64(alpha + 3.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.25 / (alpha + 3.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.25 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.0%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{1 + \beta}}{{\left(2 + \beta\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{1 + \beta}{\color{blue}{\left(\beta + 2\right)} \cdot \left(2 + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   8. +-lowering-+.f64 69.8

      \[\leadsto \frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

5. Simplified 69.8%

   \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

6. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{3 + \alpha}} \]

   2. +-lowering-+.f64 48.0

      \[\leadsto \frac{0.25}{\color{blue}{3 + \alpha}} \]

8. Simplified 48.0%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \alpha}} \]

9. Final simplification 48.0%

   \[\leadsto \frac{0.25}{\alpha + 3} \]
10. Add Preprocessing

Alternative 26: 44.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma alpha -0.027777777777777776 0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(alpha, -0.027777777777777776, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(alpha, -0.027777777777777776, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(alpha * -0.027777777777777776 + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 93.0%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around 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 69.8

      \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

5. Simplified 69.8%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \alpha} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{36} \cdot \alpha + \frac{1}{12}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\alpha \cdot \frac{-1}{36}} + \frac{1}{12} \]

   3. accelerator-lowering-fma.f64 46.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)} \]

8. Simplified 46.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.027777777777777776, 0.08333333333333333\right)} \]
9. Add Preprocessing

Alternative 27: 44.5% 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 93.0%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Add Preprocessing

3. Taylor expanded in beta around 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 69.8

      \[\leadsto \frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \color{blue}{\left(3 + \alpha\right)}} \]

5. Simplified 69.8%

   \[\leadsto \color{blue}{\frac{1 + \alpha}{\left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)\right) \cdot \left(3 + \alpha\right)}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{1}{12}} \]
7. Step-by-step derivation

8. Simplified 46.5%

   \[\leadsto \color{blue}{0.08333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))