Octave 3.8, jcobi/3

Percentage Accurate: 94.6% → 99.3%

Time: 14.0s

Alternatives: 15

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{1}{\left(\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \end{array} \]

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 1.0 / ((((beta + (2.0 + alpha)) / (1.0 + beta)) * ((alpha + (beta + 3.0)) / (1.0 + alpha))) * (alpha + (beta + 2.0)));
}

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 = 1.0d0 / ((((beta + (2.0d0 + alpha)) / (1.0d0 + beta)) * ((alpha + (beta + 3.0d0)) / (1.0d0 + alpha))) * (alpha + (beta + 2.0d0)))
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 1.0 / ((((beta + (2.0 + alpha)) / (1.0 + beta)) * ((alpha + (beta + 3.0)) / (1.0 + alpha))) * (alpha + (beta + 2.0)))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(1.0 / Float64(Float64(Float64(Float64(beta + Float64(2.0 + alpha)) / Float64(1.0 + beta)) * Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(1.0 + alpha))) * Float64(alpha + Float64(beta + 2.0))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. associate-/l/ 95.4%

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

   2. +-commutative 95.4%

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

   3. associate-+l+ 95.4%

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

   4. *-commutative 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-+l+ 95.4%

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

   7. metadata-eval 95.4%

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

   8. +-commutative 95.4%

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

   9. +-commutative 95.4%

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

   10. +-commutative 95.4%

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

   11. metadata-eval 95.4%

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

   12. metadata-eval 95.4%

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

   13. associate-+l+ 95.4%

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

3. Simplified 95.4%

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

   1. clear-num 95.4%

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

   2. inv-pow 95.4%

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

   3. *-commutative 95.4%

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

   4. associate-+r+ 95.4%

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

   5. +-commutative 95.4%

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

   6. distribute-rgt1-in 95.4%

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

   7. fma-define 95.4%

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

6. Applied egg-rr 95.4%

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

   1. unpow-1 95.4%

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

   2. associate-/r/ 95.4%

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

   3. +-commutative 95.4%

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

   4. +-commutative 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

   7. +-commutative 95.4%

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

   8. fma-undefine 95.4%

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

   9. +-commutative 95.4%

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

   10. *-commutative 95.4%

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

   11. +-commutative 95.4%

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

   12. associate-+r+ 95.4%

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

   13. distribute-lft1-in 95.4%

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

   14. +-commutative 95.4%

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

   15. +-commutative 95.4%

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

   16. +-commutative 95.4%

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

8. Simplified 95.4%

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

   1. times-frac 99.3%

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

   2. associate-+l+ 99.3%

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

   3. +-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

    1. +-commutative 99.3%

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

    2. +-commutative 99.3%

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

    3. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

    \[\leadsto \frac{1}{\left(\frac{\beta + \left(2 + \alpha\right)}{1 + \beta} \cdot \frac{\alpha + \left(\beta + 3\right)}{1 + \alpha}\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
14. Add Preprocessing

Alternative 2: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 13:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(2 + \alpha\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{3 + 2 \cdot \alpha}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 13.0)
   (/
    1.0
    (*
     (+ alpha (+ beta 2.0))
     (* (+ 2.0 alpha) (/ (+ alpha 3.0) (+ 1.0 alpha)))))
   (/
    (* (+ 1.0 alpha) (/ (- 1.0 (/ (+ 3.0 (* 2.0 alpha)) beta)) beta))
    (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 13.0) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - ((3.0 + (2.0 * alpha)) / beta)) / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 13.0d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * ((2.0d0 + alpha) * ((alpha + 3.0d0) / (1.0d0 + alpha))))
    else
        tmp = ((1.0d0 + alpha) * ((1.0d0 - ((3.0d0 + (2.0d0 * alpha)) / beta)) / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 13.0) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) * ((1.0 - ((3.0 + (2.0 * alpha)) / beta)) / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 13.0:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))))
	else:
		tmp = ((1.0 + alpha) * ((1.0 - ((3.0 + (2.0 * alpha)) / beta)) / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 13.0)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(2.0 + alpha) * Float64(Float64(alpha + 3.0) / Float64(1.0 + alpha)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(1.0 - Float64(Float64(3.0 + Float64(2.0 * alpha)) / beta)) / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 13.0)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	else
		tmp = ((1.0 + alpha) * ((1.0 - ((3.0 + (2.0 * alpha)) / beta)) / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 13.0], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(1.0 - N[(N[(3.0 + N[(2.0 * alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 13

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   if 13 < beta

   1. Initial program 89.5%

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

   3. Simplified 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. associate-/l* 76.0%

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

      3. +-commutative 76.0%

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

      4. associate-+r+ 76.0%

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

      5. associate-*r* 76.0%

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

      6. pow2 76.0%

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

      7. associate-+r+ 76.0%

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

   6. Applied egg-rr 76.0%

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

      1. *-lft-identity 76.0%

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

      2. +-commutative 76.0%

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

      3. associate-/r* 93.2%

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

      4. +-commutative 93.2%

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

      5. +-commutative 93.2%

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

      6. +-commutative 93.2%

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

      7. +-commutative 93.2%

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

   8. Simplified 93.2%

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

      1. associate-*r/ 95.4%

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

      2. div-inv 95.4%

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

      3. +-commutative 95.4%

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

      4. associate-+r+ 95.4%

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

      5. metadata-eval 95.4%

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

      6. pow-flip 96.7%

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

      7. metadata-eval 96.7%

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

      8. associate-+r+ 96.7%

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

      9. +-commutative 96.7%

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

      10. associate-+l+ 96.7%

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

      11. metadata-eval 96.7%

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

      12. +-commutative 96.7%

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

   10. Applied egg-rr 96.7%

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

   11. Taylor expanded in beta around inf 83.3%

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

       1. mul-1-neg 83.3%

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

   13. Simplified 83.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 13:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(2 + \alpha\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1 - \frac{3 + 2 \cdot \alpha}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (alpha + (beta + 3.0))));
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]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Simplified 85.6%

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

   1. times-frac 98.1%

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

   2. +-commutative 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

   \[\leadsto \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
8. Add Preprocessing

Alternative 4: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 31:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(2 + \alpha\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 31.0)
   (/
    1.0
    (*
     (+ alpha (+ beta 2.0))
     (* (+ 2.0 alpha) (/ (+ alpha 3.0) (+ 1.0 alpha)))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 31.0) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 31.0d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * ((2.0d0 + alpha) * ((alpha + 3.0d0) / (1.0d0 + alpha))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 31.0) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 31.0:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 31.0)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(2.0 + alpha) * Float64(Float64(alpha + 3.0) / Float64(1.0 + alpha)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 31.0)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 31.0], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 31

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   if 31 < beta

   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 beta around inf 84.6%

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

   4. Taylor expanded in alpha around 0 84.6%

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

      1. +-commutative 84.6%

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

      2. associate-+r+ 84.6%

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

      3. +-commutative 84.6%

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

      4. +-commutative 84.6%

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

      5. +-commutative 84.6%

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

   6. Simplified 84.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 31:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(2 + \alpha\right) \cdot \frac{\alpha + 3}{1 + \alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = 1.0 / ((2.0 + alpha) * ((2.0 + alpha) * ((alpha + 3.0) / (1.0 + alpha))))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(1.0 / Float64(Float64(2.0 + alpha) * Float64(Float64(2.0 + alpha) * Float64(Float64(alpha + 3.0) / Float64(1.0 + alpha)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(2.0 + alpha), $MachinePrecision] * N[(N[(alpha + 3.0), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in beta around 0 97.3%

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

   if 4.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-+r+ 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 92.7%

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

Alternative 6: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 + \alpha \cdot \left(2 \cdot \alpha + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   (/ 1.0 (* (+ 2.0 alpha) (+ 6.0 (* alpha (+ (* 2.0 alpha) -1.0)))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 + (alpha * ((2.0 * alpha) + -1.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.5d0) then
        tmp = 1.0d0 / ((2.0d0 + alpha) * (6.0d0 + (alpha * ((2.0d0 * alpha) + (-1.0d0)))))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = 1.0 / ((2.0 + alpha) * (6.0 + (alpha * ((2.0 * alpha) + -1.0))));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3.5:
		tmp = 1.0 / ((2.0 + alpha) * (6.0 + (alpha * ((2.0 * alpha) + -1.0))))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = Float64(1.0 / Float64(Float64(2.0 + alpha) * Float64(6.0 + Float64(alpha * Float64(Float64(2.0 * alpha) + -1.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.5)
		tmp = 1.0 / ((2.0 + alpha) * (6.0 + (alpha * ((2.0 * alpha) + -1.0))));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5], N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] * N[(6.0 + N[(alpha * N[(N[(2.0 * alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 78.0%

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

   13. Taylor expanded in beta around 0 77.9%

       \[\leadsto \frac{1}{\left(6 + \alpha \cdot \left(2 \cdot \alpha - 1\right)\right) \cdot \left(\color{blue}{2} + \alpha\right)} \]

   if 3.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-+r+ 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{1}{\left(2 + \alpha\right) \cdot \left(6 + \alpha \cdot \left(2 \cdot \alpha + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) (- 6.0 alpha)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * (6.0d0 - alpha))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6.5:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(6.0 - alpha)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * (6.0 - alpha));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 6.5], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(6.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 77.0%

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

       1. mul-1-neg 77.0%

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

   14. Simplified 77.0%

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

   if 6.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-+r+ 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.5d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.5], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 59.8%

       \[\leadsto \frac{1}{\color{blue}{6} \cdot \left(\left(\beta + 2\right) + \alpha\right)} \]

   if 5.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-+r+ 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. +-commutative 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.5)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ (+ 1.0 alpha) beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.5d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.5:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = ((1.0 + alpha) / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.5], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 59.8%

       \[\leadsto \frac{1}{\color{blue}{6} \cdot \left(\left(\beta + 2\right) + \alpha\right)} \]

   if 5.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 83.4%

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

      1. +-commutative 83.4%

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

   6. Simplified 83.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 1.0 (* (+ alpha (+ beta 2.0)) 6.0))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = 1.0d0 / ((alpha + (beta + 2.0d0)) * 6.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0)
	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 <= 5.2)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) * 6.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = 1.0 / ((alpha + (beta + 2.0)) * 6.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 59.8%

       \[\leadsto \frac{1}{\color{blue}{6} \cdot \left(\left(\beta + 2\right) + \alpha\right)} \]

   if 5.20000000000000018 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. associate-/r* 77.0%

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

      2. +-commutative 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ (/ 1.0 beta) (+ beta 3.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = 0.16666666666666666 / (beta + 2.0)
	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 <= 5.2)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 58.6%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

   if 5.20000000000000018 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 75.8%

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

      1. associate-/r* 77.0%

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

      2. +-commutative 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ 0.16666666666666666 (+ beta 2.0))
   (/ 1.0 (* beta (+ beta 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.2d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.2:
		tmp = 0.16666666666666666 / (beta + 2.0)
	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 <= 5.2)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.2)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 58.6%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

   if 5.20000000000000018 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 75.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.5) (/ 0.16666666666666666 (+ beta 2.0)) (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.5d0) then
        tmp = 0.16666666666666666d0 / (beta + 2.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.5) {
		tmp = 0.16666666666666666 / (beta + 2.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.5:
		tmp = 0.16666666666666666 / (beta + 2.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.5)
		tmp = Float64(0.16666666666666666 / Float64(beta + 2.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.5)
		tmp = 0.16666666666666666 / (beta + 2.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.5], N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt1-in 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-undefine 99.6%

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

      9. +-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+r+ 99.6%

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

      13. distribute-lft1-in 99.6%

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

      14. +-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in beta around 0 97.4%

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

       1. associate-/l* 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in alpha around 0 58.6%

       \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

   if 7.5 < beta

   1. Initial program 89.5%

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

   3. Taylor expanded in beta around inf 83.6%

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

   4. Taylor expanded in alpha around 0 75.8%

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

   5. Taylor expanded in beta around inf 75.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.16666666666666666}{\beta + 2} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.16666666666666666 (+ beta 2.0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

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 = 0.16666666666666666d0 / (beta + 2.0d0)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.16666666666666666 / (beta + 2.0);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.16666666666666666 / (beta + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.16666666666666666 / Float64(beta + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.16666666666666666 / (beta + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.16666666666666666 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

   1. associate-/l/ 95.4%

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

   2. +-commutative 95.4%

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

   3. associate-+l+ 95.4%

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

   4. *-commutative 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-+l+ 95.4%

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

   7. metadata-eval 95.4%

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

   8. +-commutative 95.4%

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

   9. +-commutative 95.4%

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

   10. +-commutative 95.4%

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

   11. metadata-eval 95.4%

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

   12. metadata-eval 95.4%

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

   13. associate-+l+ 95.4%

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

3. Simplified 95.4%

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

   1. clear-num 95.4%

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

   2. inv-pow 95.4%

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

   3. *-commutative 95.4%

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

   4. associate-+r+ 95.4%

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

   5. +-commutative 95.4%

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

   6. distribute-rgt1-in 95.4%

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

   7. fma-define 95.4%

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

6. Applied egg-rr 95.4%

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

   1. unpow-1 95.4%

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

   2. associate-/r/ 95.4%

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

   3. +-commutative 95.4%

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

   4. +-commutative 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

   7. +-commutative 95.4%

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

   8. fma-undefine 95.4%

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

   9. +-commutative 95.4%

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

   10. *-commutative 95.4%

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

   11. +-commutative 95.4%

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

   12. associate-+r+ 95.4%

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

   13. distribute-lft1-in 95.4%

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

   14. +-commutative 95.4%

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

   15. +-commutative 95.4%

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

   16. +-commutative 95.4%

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

8. Simplified 95.4%

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

9. Taylor expanded in beta around 0 71.0%

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

    1. associate-/l* 71.0%

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

11. Simplified 71.0%

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

12. Taylor expanded in alpha around 0 41.1%

    \[\leadsto \color{blue}{\frac{0.16666666666666666}{2 + \beta}} \]

13. Final simplification 41.1%

    \[\leadsto \frac{0.16666666666666666}{\beta + 2} \]
14. Add Preprocessing

Alternative 15: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.3333333333333333}{\beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.3333333333333333 beta))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.3333333333333333d0 / beta
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.3333333333333333 / beta;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.3333333333333333 / beta

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.3333333333333333 / beta)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.3333333333333333 / beta;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.3333333333333333 / beta), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in beta around inf 30.1%

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

4. Taylor expanded in alpha around 0 27.5%

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

5. Taylor expanded in beta around 0 4.3%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\beta}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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)))